1
2y² − 3y + 4 is —
💡 Hint:
The highest power of y is 2.
Explanation:
Degree 2, so quadratic.
Class 10 · Maths · Chapter 2 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Polynomials
Polynomials
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- 1 Degree: linear, quadratic, cubic
- 2 The graph of a linear polynomial and its one zero
- 3 The quadratic parabola — Exercise 2.1
- 4 Zeroes and coefficients — question 1 of Exercise 2.2
- 5 A polynomial from the sum and the product — question 2 of Exercise 2.2
- 6 The division algorithm — Exercise 2.3
- 7 Zeroes of a cubic — optional Exercise 2.4
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
Degree: linear, quadratic, cubic
घात: रैखिक, द्विघात, त्रिघात · NCERT 2.1 · degree
The highest power decides the nameNotes
The highest power of x in p(x) is the degree. 4x + 2 has degree 1, so it is linear. 2y² − 3y + 4 has degree 2, so it is quadratic. 5x³ − 4x² + x − 2 has degree 3, so it is cubic.
A quadratic has the form ax² + bx + c with a ≠ 0. If a is zero, the x² term drops and the degree is no longer 2.
The opening of the book — name the degreeActivity
Write the degree and the name of each: 4x + 2, 2y² − 3y + 4, 5x³ − 4x² + x − 2, and a term whose variable has power 6.
1/x and √x are not polynomials. A degree is not negative and not a fraction.
Worked exampleExample
Question: What is the degree of 7u⁶ − (4/3)u⁴ + (1/2)u − 8?
Rule: degree = the highest power of the variable.
Substitution: the powers are 6, 4, 1 and a constant term. The greatest is 6.
Answer: the degree is 6. It is not linear, not quadratic and not cubic.
10-second revision
- Degree 1 linear, 2 quadratic, 3 cubic
- In a quadratic, a ≠ 0
- 1/x is not a polynomial
Board tip · BSEBBoard tip
When you write the name, write the degree too. Only “polynomial” stays incomplete.
Board tip · CBSEBoard tip
Treat the constant term as degree 0. It is not the highest power when a variable term is present.
Check your understandingall correct = mastery ★
2
The degree of 4x + 2 is 2 because the digit 2 is written in it.
Explanation:
False — the degree is the power of the variable. Here the power of x is 1, so it is linear.
3
The degree of 5x³ − 4x² + x − 2 is ______.
💡 Hint:
Look at the highest power.
Explanation:
3. It is cubic.
4
Write the general quadratic and say the condition on a.
Model answer (board format)
ax² + bx + c, where a, b, c are real and a ≠ 0.
Check your answer:
The graph of a linear polynomial and its one zero
रैखिक बहुपद का ग्राफ और एक शून्यक · NCERT 2.2 · a linear graph
The meeting with the x-axis is the zeroNotes
A zero of p(x) is an x for which p(x) = 0. On the graph it is the x-coordinate of the point where y = p(x) meets the x-axis.
The zero of the linear polynomial ax + b is −b/a. A straight line cuts the x-axis at one point, so there is one zero.
A line from a table of valuesActivity
Write a few values of y = 2x + 4 and see where the line cuts the x-axis. At that x, y must be zero.
The same method is used later for the graphs in Exercise 2.1: count the meetings with the x-axis.
| x | −2 | −1 | 0 | 1 |
|---|---|---|---|---|
| y = 2x + 4 | 0 | 2 | 4 | 6 |
Worked exampleExample
Question: Find the zero of 2x + 4 and match it with the graph.
Formula: the zero of ax + b is −b/a. Here a = 2 and b = 4.
Substitution: −b/a = −4/2 = −2. Check: 2(−2) + 4 = 0.
In the table, y = 0 when x = −2. The line cuts the x-axis at (−2, 0). The answer is −2.
10-second revision
- A zero = the x of a crossing on the x-axis
- The zero of ax + b is −b/a
- A linear graph cuts once
Board tip · BSEBBoard tip
After the formula, write one line showing p(zero) = 0.
Board tip · CBSEBoard tip
The crossing on the y-axis is not a zero. A zero is the point on the x-axis.
Check your understandingall correct = mastery ★
1
The zero of 3x − 6 is —
💡 Hint:
−b/a = −(−6)/3.
Explanation:
−(−6)/3 = 2. Check: 3×2 − 6 = 0.
2
The graph of a linear polynomial cuts the x-axis at two points.
Explanation:
False — a straight line cuts the x-axis at one point.
3
The zero of ax + b is ______.
💡 Hint:
Divide by a and keep the minus sign.
Explanation:
−b/a.
4
In the table of y = 2x + 4 the zero is —
💡 Hint:
Where y = 0.
Explanation:
At x = −2, y = 0.
5
Find the zero of 5x + 10 by putting it in the formula, and show the check.
Model answer (board format)
The zero = −b/a = −10/5 = −2. Check: 5(−2) + 10 = 0.
Check your answer:
The quadratic parabola — Exercise 2.1
द्विघात का परवलय — अभ्यास 2.1 · NCERT 2.2 · Exercise 2.1
Count the crossings, at most twoNotes
The graph of a quadratic is a parabola. It may cut the x-axis at two points, touch it at one point, or miss it. So there may be 0, 1 or 2 real zeroes. At most two.
If a > 0 the parabola opens upward. Between the two zeroes the curve stays below the x-axis, as in the figure on this page.
Exercise 2.1 — the graphs in Fig. 2.10Activity
Exercise 2.1 has one question. For each graph of y = p(x) in Fig. 2.10, write the number of zeroes.
Method: the number of meetings with the x-axis is the number of zeroes. A touch is one zero. No meeting means no zero.
Worked exampleExample
Question: How many zeroes does the graph of y = (x − 1)(x − 4) have, and what are they?
Rule: a zero is where y = 0, that is a crossing on the x-axis.
Substitution: (x − 1)(x − 4) = 0 gives x = 1 or x = 4.
There are two different crossings. The answer is two zeroes, 1 and 4. The parabola in the figure cuts in the same way, at two places.
10-second revision
- Number of zeroes = meetings with the x-axis
- A quadratic has 0, 1 or 2 real zeroes
- A touch counts as one zero
Board tip · BSEBBoard tip
Write a separate count for each picture in Fig. 2.10. Do not paste one answer onto all of them.
Board tip · CBSEBoard tip
“A quadratic has exactly two zeroes” is the wrong sentence. The right sentence is “at most two”.
Check your understandingall correct = mastery ★
1
A quadratic polynomial can have this many real zeroes —
💡 Hint:
The parabola can also miss.
Explanation:
0, 1 or 2. At most two.
2
A graph touching the x-axis once is one zero.
Explanation:
True — a touch is also a meeting, so it counts as one zero.
3
The zeroes of y = p(x) are the ______-coordinates of the points where it meets the x-axis.
💡 Hint:
Not y.
Explanation:
The x-coordinates.
4
If a parabola does not meet the x-axis at all, the number of zeroes is —
💡 Hint:
No meeting means no zero.
Explanation:
0.
5
Write the method of Exercise 2.1 in one sentence. Give a quadratic that has two zeroes.
Model answer (board format)
The number of meetings with the x-axis is the number of zeroes. y = (x − 1)(x − 4) has zeroes 1 and 4, so two.
Check your answer:
6
Between the two zeroes of a parabola with a > 0, the curve lies —
💡 Hint:
An upward-opening parabola dips in the middle.
Explanation:
Below the x-axis. That is what the figure shows.
Zeroes and coefficients — question 1 of Exercise 2.2
शून्यक और गुणांक — अभ्यास 2.2 का पहला प्रश्न · NCERT 2.3 · Exercise 2.2 question 1
The sum is −b/a, the product is c/aNotes
If α and β are the zeroes of ax² + bx + c, with a ≠ 0, then
α + β = −b/a and αβ = c/a
The book shows this on x² + 7x + 10. The factors (x + 2)(x + 5) give zeroes −2 and −5. The sum −7 and the product 10 match −b/a and c/a.
Exercise 2.2 — the six expressions in question 1Activity
Find the zeroes of each and match the sum and the product with the coefficients:
(i) x² − 2x − 8 (ii) 4s² − 4s + 1 (iii) 6x² − 7x − 3
(iv) 4u² + 8u (v) t² − 15 (vi) 3x² − x − 4
Worked exampleExample
Question: Zeroes of x² − 2x − 8 and the link.
Factors: x² − 2x − 8 = (x − 4)(x + 2). Zeroes 4 and −2.
Formula: sum = −b/a, product = c/a. Here a = 1, b = −2, c = −8.
Substitution: −b/a = −(−2)/1 = 2. Sum of zeroes 4 + (−2) = 2.
c/a = −8/1 = −8. Product 4×(−2) = −8. The link matches.
10-second revision
- α + β = −b/a
- αβ = c/a
- The zeroes of x² − 2x − 8 are 4 and −2
Board tip · BSEBBoard tip
The minus sign sits in the sum. If b is already negative, −b becomes positive.
Board tip · CBSEBoard tip
4s² − 4s + 1 has the zero 1/2 twice. Do not forget to write the sum 1 and the product 1/4.
Check your understandingall correct = mastery ★
1
The zeroes of x² + 7x + 10 are —
💡 Hint:
(x + 2)(x + 5).
Explanation:
−2 and −5. Sum −7 = −b/a, product 10 = c/a.
2
The sum of the zeroes of t² − 15 is 0.
Explanation:
True — the zeroes are √15 and −√15, and the sum is 0 = −b/a, because b = 0.
3
In ax² + bx + c the product of the zeroes is ______.
💡 Hint:
The sum has the minus, the product does not.
Explanation:
c/a.
4
The zeroes of 4u² + 8u are —
💡 Hint:
4u(u + 2).
Explanation:
0 and −2. Sum −2 = −8/4. Product 0.
5
Find the zeroes of 3x² − x − 4. Match the sum and the product with −b/a and c/a.
Model answer (board format)
From (x + 1)(3x − 4) = 0 the zeroes are −1 and 4/3. Sum = −1 + 4/3 = 1/3 = −(−1)/3. Product = −4/3 = c/a.
Check your answer:
A polynomial from the sum and the product — question 2 of Exercise 2.2
योग और गुणनफल से बहुपद — अभ्यास 2.2 का दूसरा प्रश्न · Exercise 2.2 · question 2
With a = 1 the polynomial is x² − (sum)x + productNotes
If the sum S and the product P are given, one quadratic is x² − Sx + P. Any other answer has the form k(x² − Sx + P), with k ≠ 0.
The book’s example: the polynomial with sum −3 and product 2 is x² + 3x + 2, because −S = 3.
Exercise 2.2 — question 2Activity
Write one quadratic for each pair of sum and product:
(i) 1/4 and −1 (ii) √2 and 1/3 (iii) 0 and √5
(iv) 1 and 1 (v) −1/4 and 1/4 (vi) 4 and 1
Worked exampleExample
Question: A quadratic with sum −3 and product 2.
Formula: x² − (sum)x + product. Also α + β = −b/a and αβ = c/a.
Substitution: take a = 1. −b/a = −3, so b = 3. c/a = 2, so c = 2.
Answer: x² + 3x + 2. Any k(x² + 3x + 2) is also correct.
10-second revision
- x² − (sum)x + product
- Multiplying by k keeps the same zeroes
- Sum −3, product 2 → x² + 3x + 2
Board tip · BSEBBoard tip
If the sum is negative, the middle term becomes positive. Write the sign once and check it.
Board tip · CBSEBoard tip
If the sum is a fraction you may multiply through by the denominator. For 1/4 and −1, 4x² − x − 4 is also correct.
Check your understandingall correct = mastery ★
1
One polynomial with sum 4 and product 1 is —
💡 Hint:
The middle term is minus the sum.
Explanation:
x² − 4x + 1.
2
The zeroes of 2(x² + 3x + 2) become different from those of x² + 3x + 2.
Explanation:
False — multiplying by a non-zero k keeps the zeroes the same.
3
One polynomial with sum 0 and product √5 is x² + ______.
💡 Hint:
The middle term is 0.
Explanation:
x² + √5.
4
Write a quadratic with sum 1/4 and product −1. Show both the a = 1 form and the a = 4 form.
Model answer (board format)
x² − (1/4)x − 1. Multiplying through by 4 gives 4x² − x − 4. Both have the same zeroes.
Check your answer:
The division algorithm — Exercise 2.3
विभाजन एल्गोरिथ्म — अभ्यास 2.3 · NCERT 2.4 · Exercise 2.3
Dividend = divisor × quotient + remainderNotes
For p(x) and a non-zero g(x) there are q(x) and r(x) such that
p(x) = g(x) q(x) + r(x)
where r(x) = 0, or the degree of r is less than the degree of g. Write the terms in falling powers. That is standard form. Divide by rewriting 1 + 2x + x² as x² + 2x + 1.
In the Bihar book this is Exercise 2.3. The NCERT reprint of 2026-27 does not print this section.
Exercise 2.3 — all five questionsActivity
Question 1: find quotient and remainder. (i) x³ − 3x² + 5x − 3 by x² − 2. (ii) x⁴ − 3x² + 4x + 5 by x² − x + 1. (iii) x⁴ − 5x + 6 by 2 − x².
Question 2: see by division whether the first polynomial is a factor of the second. Question 3: two zeroes of 3x⁴ + 6x³ − 2x² − 10x − 5 are √(5/3) and −√(5/3); find the others.
Question 4: dividing x³ − 3x² + x + 2 by g(x) gave quotient x − 2 and remainder −2x + 4. Find g(x). Question 5: give examples where the degrees match, or the remainder has degree 0.
Worked exampleExample
Question: Divide x³ − 3x² + 5x − 3 by x² − 2.
Formula: p = gq + r, and stop when the degree of r is less than 2.
Step 1: x³/x² = x. x(x² − 2) = x³ − 2x. Subtraction leaves −3x² + 7x − 3.
Step 2: −3x²/x² = −3. −3(x² − 2) = −3x² + 6. Subtraction leaves 7x − 9.
Quotient x − 3, remainder 7x − 9. Check: (x² − 2)(x − 3) + (7x − 9) = x³ − 3x² + 5x − 3. Answer q = x − 3, r = 7x − 9.
10-second revision
- p = gq + r
- The remainder is 0 or its degree is less than the divisor
- Standard form is falling powers
Board tip · BSEBBoard tip
At the end expand gq + r and match it with p. That check saves the marks.
Board tip · CBSEBoard tip
In question 4, g = (p − r)/q. Forgetting to subtract the remainder changes the divisor.
Check your understandingall correct = mastery ★
1
On dividing x³ − 3x² + 5x − 3 by x² − 2, the quotient is —
💡 Hint:
7x − 9 is the remainder, not the quotient.
Explanation:
The quotient is x − 3 and the remainder is 7x − 9.
2
Division stops when the degree of the remainder is less than the degree of the divisor, or the remainder is zero.
Explanation:
True — that is the stopping condition of the algorithm.
3
Division algorithm: p(x) = g(x) q(x) + ______.
💡 Hint:
The last term is the remainder.
Explanation:
r(x).
4
If x³ − 3x² + x + 2 = g(x)(x − 2) + (−2x + 4), find g(x).
Model answer (board format)
g = (p − r)/(x − 2). p − r = x³ − 3x² + 3x − 2. Division by x − 2 gives x² − x + 1. Check: (x² − x + 1)(x − 2) + (−2x + 4) = x³ − 3x² + x + 2.
Check your answer:
5
The other two zeroes of 3x⁴ + 6x³ − 2x² − 10x − 5, if two zeroes are ±√(5/3), are —
💡 Hint:
Divide by 3x² − 5. The quotient is (x + 1)².
Explanation:
The quotient is x² + 2x + 1 = (x + 1)², so the zeroes are −1, −1.
6
An example of deg p = deg q arises when the divisor is a constant.
Explanation:
True — for example p = 2x + 2, g = 2, q = x + 1, r = 0. Both degrees are 1.
Zeroes of a cubic — optional Exercise 2.4
त्रिघात के शून्यक — वैकल्पिक अभ्यास 2.4 · Optional Exercise 2.4 · not for the examination
Three links, and this exercise is not for the examinationNotes
If α, β, γ are the zeroes of ax³ + bx² + cx + d, then
α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.
The book stars Exercise 2.4: these questions are not for the examination. The links are still in the summary. A cubic has at most three zeroes.
Exercise 2.4 — the optional five questionsActivity
Question 1: check 2x³ + x² − 5x + 2 with 1/2, 1, −2, and x³ − 4x² + 5x − 2 with 2, 1, 1. Check the zeroes and the three links.
Question 2: a cubic with sum 2, sum of products two at a time −7, and product −14. Question 3: if the zeroes of x³ − 3x² + x + 1 are a − b, a, a + b, find a and b.
Questions 4 and 5 continue factoring and division. If 2 ± √3 are two zeroes, find the others, and if division by x² − 2x + k leaves remainder x + a, find k and a.
Worked exampleExample
Question: A cubic with sum 2, sum of products two at a time −7, and product −14.
Formula: −b/a = 2, c/a = −7, −d/a = −14.
Substitution: take a = 1. Then b = −2, c = −7, d = 14.
Answer: x³ − 2x² − 7x + 14.
10-second revision
- Sum −b/a, pair-sum c/a, product −d/a
- Exercise 2.4 is not for the examination
- x³ − 2x² − 7x + 14 fits those three numbers
Board tip · BSEBBoard tip
The product carries a minus: αβγ = −d/a. The sum also carries a minus. The middle link does not.
Board tip · CBSEBoard tip
Optional does not mean the link is wrong. Line 6 of the summary is the same link.
Check your understandingall correct = mastery ★
1
The sum of the zeroes 1/2, 1, −2 of 2x³ + x² − 5x + 2 is —
💡 Hint:
−b/a = −1/2.
Explanation:
1/2 + 1 − 2 = −1/2 = −b/a.
2
The product of the zeroes of a cubic is −d/a.
Explanation:
True — αβγ = −d/a. 1/2 × 1 × (−2) = −1 and −d/a = −2/2 = −1.
3
One cubic with sum 2, pair-sum −7 and product −14 is x³ − 2x² − 7x + ______.
💡 Hint:
−d/a = −14 and a = 1.
Explanation:
14.
4
The zeroes of x³ − 3x² + x + 1 are a − b, a and a + b. Find a and b.
Model answer (board format)
The sum 3a equals 3, so a = 1. The product a(a² − b²) = −d/a = −1. 1 − b² = −1, so b² = 2 and b = ±√2.
Check your answer:
5
About Exercise 2.4 the book says —
💡 Hint:
The starred line.
Explanation:
The book writes that these questions are not from the examination point of view.
❓ Full question bank — with answers and explanations — 65 questions
Pick a type. The 35 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA120 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option.
1
The degree of 4x + 2 is —
💡 Hint:
Look at the power of x.
Explanation:
1, linear.
2
In the quadratic ax² + bx + c the condition is —
💡 Hint:
The x² term has to stay.
Explanation:
a ≠ 0.
3
The zero of 2x + 4 is —
💡 Hint:
−b/a.
Explanation:
−2.
4
If the graph cuts the x-axis twice, the zeroes are —
💡 Hint:
Count the meetings.
Explanation:
2.
5
The sum of the zeroes of x² + 7x + 10 is —
💡 Hint:
−b/a.
Explanation:
−7.
6
The zeroes of x² − 2x − 8 are —
💡 Hint:
(x − 4)(x + 2).
Explanation:
4 and −2.
7
A polynomial with sum −3 and product 2 is —
💡 Hint:
x² − (sum)x + product.
Explanation:
x² + 3x + 2.
8
The stopping condition in division is —
💡 Hint:
The line of the algorithm.
Explanation:
The remainder is 0 or its degree is smaller.
9
On dividing x³ − 3x² + 5x − 3 by x² − 2, the remainder is —
💡 Hint:
The quotient is x − 3.
Explanation:
7x − 9.
10
The sum of the zeroes of a cubic is —
💡 Hint:
α + β + γ.
Explanation:
−b/a.
11
The zero of 4s² − 4s + 1 is —
💡 Hint:
(2s − 1)².
Explanation:
1/2 twice.
12
Exercise 2.4 is —
💡 Hint:
The star.
Explanation:
It is optional and not for the examination.
True or false
0/81
1/x is a polynomial.
Explanation:
False — a degree is not negative.
2
A quadratic has at most two zeroes.
Explanation:
True.
3
The crossing on the y-axis is a zero of the polynomial.
Explanation:
False — a zero is an x on the x-axis.
4
The product of the zeroes of x² + 7x + 10 is 10.
Explanation:
True — c/a = 10.
5
The zeroes of k(x² + 3x + 2), k ≠ 0, change.
Explanation:
False — the zeroes stay the same.
6
Division is treated as finished even when the degree of the remainder is greater than the degree of the divisor.
Explanation:
False — you still have to divide further.
7
αβγ = −d/a is the product link for a cubic.
Explanation:
True.
8
A parabola missing the x-axis counts as two zeroes.
Explanation:
False — no meeting means 0 zeroes.
Fill in the blanks
0/81
The degree of 2y² − 3y + 4 is ______.
💡 Hint:
2.
Explanation:
The highest power.
2
The zero of the linear ax + b is ______.
💡 Hint:
−b/a.
Explanation:
Minus b over a.
3
The sum of the zeroes is −b/a and the product is ______.
💡 Hint:
c/a.
Explanation:
The quadratic.
4
The polynomial with sum 4 and product 1 is x² − ______ x + 1.
💡 Hint:
4.
Explanation:
The middle term is the sum with a minus.
5
p(x) = g(x) q(x) + ______.
💡 Hint:
r(x).
Explanation:
The remainder.
6
Dividing x³ − 3x² + 5x − 3 by x² − 2 gives quotient x − ______.
💡 Hint:
3.
Explanation:
The remainder 7x − 9 is separate.
7
For a cubic, αβ + βγ + γα = ______.
💡 Hint:
c/a.
Explanation:
The middle link.
8
One zero of 4u² + 8u is ______.
💡 Hint:
0 or −2.
Explanation:
4u(u + 2).
Match
0/21
Match the degree with the name.
Column B: A. Cubic · B. A crossing on the x-axis · C. Linear · D. Quadratic
1. Degree 1
2. Degree 2
3. Degree 3
4. A zero
Explanation:
1 linear, 2 quadratic, 3 cubic, and a zero is a crossing on the x-axis.
2
Match the link with its formula.
Column B: A. c/a · B. −d/a · C. p = gq + r · D. −b/a
1. Quadratic sum
2. Quadratic product
3. Cubic product
4. Division
Explanation:
Sum −b/a, product c/a, cubic product −d/a, division p = gq + r.
Assertion–reason
0/5Check both statements first.
1
Assertion (A): The zeroes of x² + 7x + 10 are −2 and −5.
Reason (R): x² + 7x + 10 = (x + 2)(x + 5).
Explanation:
Both are true and R explains A.
2
Assertion (A): A quadratic has at most two zeroes.
Reason (R): Dividend = divisor × quotient + remainder.
Explanation:
Both are true, but R does not explain A.
3
Assertion (A): The zero of 2x + 4 is −2.
Reason (R): This zero is the crossing on the y-axis.
Explanation:
A is true. R is false — the zero is on the x-axis.
4
Assertion (A): The product of the zeroes of a cubic is d/a.
Reason (R): αβγ = −d/a.
Explanation:
A is false because the minus sign is missing. R is true.
5
Assertion (A): x³ − 3x² + 5x − 3 = (x² − 2)(x − 3) + (7x − 9).
Reason (R): The way to check a division is to show that gq + r equals p.
Explanation:
Both are true and R says why A is the check.
Coefficient practice
0/4This is coefficient practice. It is not a result of this maths chapter. These are small equations of water and carbon dioxide forming.
1
This is coefficient practice and not a result of this maths chapter. Balance the equation of water forming. Write coefficients only in the coefficient places.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. Balance the equation of carbon dioxide forming from carbon.
C + O2 → CO2
Explanation: C + O2 → CO2
3
This is coefficient practice and not a result of this maths chapter. Balance the equation of carbon dioxide forming from carbon monoxide.
CO + O2 → CO2
Explanation: 2CO + O2 → 2CO2
4
This is coefficient practice and not a result of this maths chapter. Burning methane forms carbon dioxide and water. Balance the equation.
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each polynomial by the name of its degree.
4x + 2
2y² − 3y + 4
5x³ − 4x² + x − 2
x² − 2x − 8
Explanation:
4x+2 is linear, both with a square are quadratic, and the one with x³ is cubic.
2
Place each graph description with the number of zeroes.
A line cuts the x-axis once
A parabola cuts at two points
A parabola misses the x-axis
A parabola touches at one point
Explanation:
One cut or one touch = 1. Two cuts = 2. No meeting = 0.
Very short answer
0/51
Define the degree of a polynomial.
Model answer (board format)
In a polynomial in one variable, the highest power of that variable is the degree.
Check your answer:
2
Write the link between a zero and the graph in one line.
Model answer (board format)
The zeroes are the x-coordinates of the points where y = p(x) meets the x-axis.
Check your answer:
3
Write the sum and the product of the zeroes of a quadratic.
Model answer (board format)
α + β = −b/a and αβ = c/a.
Check your answer:
4
Write the identity of the division algorithm.
Model answer (board format)
p(x) = g(x) q(x) + r(x), where r = 0 or deg r is less than deg g.
Check your answer:
5
Write one quadratic with sum 1 and product 1.
Model answer (board format)
x² − x + 1.
Check your answer:
Short answer
0/41
Find the zeroes of 6x² − 7x − 3 and match them with −b/a and c/a.
Model answer (board format)
From (3x + 1)(2x − 3) = 0 the zeroes are −1/3 and 3/2. Sum = 7/6 = −(−7)/6. Product = −1/2 = −3/6.
Check your answer:
2
Divide x⁴ − 3x² + 4x + 5 by x² − x + 1, write the quotient and the remainder, and give one check line.
Model answer (board format)
Quotient x² + x − 3 and remainder 8. Check: (x² − x + 1)(x² + x − 3) + 8 = x⁴ − 3x² + 4x + 5.
Check your answer:
3
The zeroes of 2x³ + x² − 5x + 2 are 1/2, 1 and −2. Check all three links.
Model answer (board format)
Sum = −1/2 = −b/a. Pair-sum = 1/2 − 2 − 1 = −5/2 = c/a. Product = −1 = −d/a.
Check your answer:
4
Write p, g, q, r so that the remainder has degree 0. Check with the identity.
Model answer (board format)
p(x) = x² + 1, g(x) = x, q(x) = x, r(x) = 1. x·x + 1 = x² + 1, and the remainder has degree 0.
Check your answer:
Long answer
0/31
Find the zeroes of x² − 2x − 8, match the sum and the product, and write the general form with k that has the same sum and product.
Model answer (board format)
The zeroes are 4 and −2. Sum 2 = −(−2)/1. Product −8 = −8/1. The general form is k(x² − 2x − 8), k ≠ 0.
Check your answer:
2
Write the division algorithm. Divide x³ − 3x² + 5x − 3 by x² − 2 and show the check of gq + r in full steps.
Model answer (board format)
p = gq + r, with r = 0 or deg r smaller. Subtracting x(x² − 2) leaves −3x² + 7x − 3. Subtracting −3(x² − 2) leaves remainder 7x − 9. Quotient x − 3. (x² − 2)(x − 3) + 7x − 9 = x³ − 3x² − 2x + 6 + 7x − 9 = x³ − 3x² + 5x − 3.
Check your answer:
3
A parabola cuts the x-axis at two zeroes. Using the figure, say what is counted as a zero. Then, taking 1/2, 1 and −2 as the zeroes of 2x³ + x² − 5x + 2, check the product link.
Model answer (board format)
The zeroes are the x-coordinates of the crossings on the x-axis. Two crossings mean two zeroes. The product (1/2)×1×(−2) = −1 and −d/a = −2/2 = −1, so the link matches.
Check your answer:
BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
The zeroes of t² − 15 are —
💡 Hint:
a² − b².
Explanation:
√15 and −√15.
2
The zero of 3x − 6 is —
💡 Hint:
−b/a.
Explanation:
2.
3
α + β = ______ / a. Write the numerator, including the minus sign.
💡 Hint:
−b.
Explanation:
The sum.
4
Write the zeroes of 4s² − 4s + 1 and the sum and product.
Model answer (board format)
The zeroes are 1/2 and 1/2. Sum 1 = 4/4. Product 1/4 = 1/4.
Check your answer:
5
This is coefficient practice and not a result of this maths chapter. Balance the equation of water forming.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
6
Write the division algorithm and divide x³ − 3x² + 5x − 3 by x² − 2, showing the check. Say why Exercise 2.4 is marked apart from the examination.
Model answer (board format)
p = gq + r, with the degree of the remainder less than the divisor. Quotient x − 3, remainder 7x − 9. (x² − 2)(x − 3) + 7x − 9 is the original polynomial. On Exercise 2.4 the book writes that these questions are not for the examination.
Check your answer:
CBSE-style questions
0/6These are competency-based practice questions. They are not copies of a past paper.
1
A student says a quadratic has exactly two zeroes, while the parabola stays above the x-axis. The correct reading is —
💡 Hint:
Count the meetings.
Explanation:
No meeting, so 0 real zeroes. At most two does not mean exactly two.
2
A shop’s profit is tied to the polynomial x² − 5x + 6. For which x is the profit zero?
💡 Hint:
(x − 2)(x − 3).
Explanation:
x = 2 and x = 3. Sum 5 = −b/a.
3
Assertion (A): The sum of the zeroes of 4x² − x − 4 is 1/4.
Reason (R): The product of the zeroes of this same polynomial is −1.
Explanation:
A is true, −b/a = 1/4. R is also true, c/a = −1. R does not explain the sum.
4
Assertion (A): The zeroes of x² − 5x + 6 are 2 and 3.
Reason (R): x² − 5x + 6 = (x − 2)(x − 3).
Explanation:
Both are true and R explains A.
5
A student finds remainder 7x − 9 and says that x² − 2 is a factor of x³ − 3x² + 5x − 3. Where is the slip? Write the correct quotient and remainder.
Model answer (board format)
It is a factor only when the remainder is 0. Here the remainder is 7x − 9, so x² − 2 is not a factor. The quotient is x − 3.
Check your answer:
6
Two path lengths are like the zeroes of a polynomial: sum 4 and product 1. Write one polynomial and say why multiplying by k does not change the paths.
Model answer (board format)
x² − 4x + 1. The zeroes of k(x² − 4x + 1) stay the same, because a non-zero k does not change the roots of the equation.
Check your answer:
🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
🏛️ Model questions on one page →
The verified label will be used only when a source page for the question is available.
CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
Wrong questions return soon; correct ones return after a few days.
🧠 What you learned + equation sheet
What you learned
- Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic.
- A zero is an x where the graph meets the x-axis.
- A quadratic has at most two zeroes, a cubic at most three.
- For ax² + bx + c the sum is −b/a and the product is c/a.
- For a cubic the sum is −b/a, the sum of products two at a time is c/a, and the product is −d/a.
- p(x) = g(x)q(x) + r(x), and the degree of the remainder is less than the degree of the divisor.
| What | Keep this |
|---|---|
| Linear zero | −b/a |
| Sum | α + β = −b/a |
| Product | αβ = c/a |
| Cubic sum | α + β + γ = −b/a |
| Sum in pairs | αβ + βγ + γα = c/a |
| Division | p = gq + r |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.