CBSE X Maths — Polynomials: Notes, Practice Questions & Worked Solutions

CBSE Class X Mathematics (041/241) · Unit II: Algebra

Polynomials

A polynomial’s zeros are the values that make it equal to zero. For a quadratic, there’s a direct link between its zeros and its coefficients — no need to solve the equation to know their sum and product.

1. Zeros of a Polynomial

A zero of a polynomial p(x) is a value of x for which p(x) = 0. Graphically, the zeros of a polynomial are the x-coordinates of the points where its graph crosses the x-axis.

Worked example — finding zeros by factorisation Find the zeros of x² − 3x − 10.
Factorise: x² − 3x − 10 = (x − 5)(x + 2)
Setting each factor to zero: x = 5 or x = −2. Zeros are 5 and −2.

2. Relationship Between Zeros and Coefficients

For a quadratic polynomial ax² + bx + c with zeros α and β:

α + β = −b/a

α × β = c/a

Worked example — verifying the relationship Verify the relationship between zeros and coefficients for x² + 7x + 12.
Factorise: (x + 3)(x + 4) → zeros are −3 and −4.
Sum: −3 + (−4) = −7, and −b/a = −7/1 = −7 ✓
Product: (−3)(−4) = 12, and c/a = 12/1 = 12 ✓

Worked example — building a polynomial from its zeros Find a quadratic polynomial whose zeros are 3 and −1.
Sum = 3 + (−1) = 2. Product = 3 × (−1) = −3.
A quadratic with these zeros is x² − (sum)x + (product) = x² − 2x − 3.

Worked example — using the relationship without finding the roots If α, β are zeros of 2x² − 5x + 3, find α² + β² without solving for α and β directly.
Sum = −b/a = 5/2. Product = c/a = 3/2.
α² + β² = (α+β)² − 2αβ = (5/2)² − 2(3/2) = 25/4 − 3 = 13/4

Common mistakes to watch for

Forgetting the negative sign: the sum of zeros is −b/a, not b/a.
Applying the sum/product formulas to a polynomial that isn’t in the standard form ax²+bx+c=0 — rearrange first.
When a quadratic has no x term (like 4x² − 9), remembering that b = 0, so the sum of zeros is 0.

Key Vocabulary Recap

Zero of a polynomial — a value of x for which p(x) = 0

Quadratic polynomial — a polynomial of degree 2, ax²+bx+c

Syllabus Reference

This note covers CBSE Class X Mathematics Unit II, Polynomials: zeros of a polynomial, and the relationship between zeros and coefficients of quadratic polynomials.

Practice questions

CBSE Class X Mathematics · Unit II: Polynomials — Practice Question Bank (no answers)

Polynomials — Practice

Covers: finding zeros by factorisation, and the relationship between zeros and coefficients of quadratics. 10 questions.

Find the zeros of x² − 3x − 10 by factorisation.

Verify the relationship between zeros and coefficients for x² + 7x + 12.

Find a quadratic polynomial whose zeros are 3 and −1.

If one zero of x² − 4x + k is 3, find the value of k.

Find the sum and product of the zeros of 4x² − 9.

Find the zeros of 6x² − 7x − 3 by factorisation.

Find a quadratic polynomial whose sum of zeros is −3 and product is 2.

If α, β are the zeros of 2x² − 5x + 3, find α² + β² without finding α and β individually.

Explain how the graph of a quadratic polynomial relates to the number of its real zeros.

Q10

Find the zeros of x² − 2x − 8 and verify the relationship between the zeros and the coefficients.

More Practice

Q11

State the maximum number of zeros a quadratic polynomial can have, with reasoning.

Q12

Find the zeros of x² − 15 and verify the sum/product relationship.

Q13

If the zeros of x² + (a+1)x + b are 2 and −3, find a and b.

Q14

Find a quadratic polynomial whose zeros are 2+√3 and 2−√3.

Q15

If α and β are the zeros of x² − 5x + 6, find 1/α + 1/β.

Q16

Find the zeros of 2x² + 3x − 2 by factorisation.

Q17

If one zero of the polynomial 3x² − 8x + (2k+1) is seven times the other, find k.

Q18

Find the zeros of 4x² − 4x + 1, and comment on whether they are equal or distinct.

Q19

If the sum of the zeros of kx² + 3x + 2k equals the product of the zeros, find k.

Q20

Find a polynomial whose zeros are the reciprocals of the zeros of 2x² − 3x + 1.

Worked solutions

Show worked solutions

CBSE Class X Mathematics · Unit II: Polynomials — Solutions

Polynomials — Practice: Solutions

Zeros of x²−3x−10.

Factorise

x²−3x−10=(x−5)(x+2)

✓x = 5 or x = −2

Verify relationship for x²+7x+12.

Factorise

(x+3)(x+4) → zeros −3, −4

✓Sum: −3+(−4)=−7=−b/a ✓ Product: (−3)(−4)=12=c/a ✓

Quadratic with zeros 3 and −1.

Sum = 2, Product = −3

x² − (sum)x + (product)

✓x² − 2x − 3

One zero of x²−4x+k is 3. Find k.

Substitute x=3, set equal to 0

9−12+k=0

✓k = 3

Sum and product of zeros of 4x²−9.

Here a=4, b=0, c=−9

sum=−b/a=0 product=c/a=−9/4

✓Sum = 0, Product = −9/4

Zeros of 6x²−7x−3.

Split the middle term

6x²−9x+2x−3=3x(2x−3)+1(2x−3)=(2x−3)(3x+1)

✓x = 3/2 or x = −1/3

Sum=−3, product=2. Find quadratic.

✓x² + 3x + 2

α²+β² for 2x²−5x+3.

Sum=5/2, Product=3/2

α²+β²=(sum)²−2(product)

Substitute

(5/2)²−2(3/2)=25/4−3

✓13/4

Graph vs number of real zeros.

✓The zeros of a quadratic polynomial are the x-coordinates of the points where its parabola crosses the x-axis. If the parabola crosses the x-axis at two points, there are two distinct real zeros; if it just touches the axis at one point, there is one repeated zero; if it doesn’t meet the x-axis at all, there are no real zeros.

Q10

Zeros of x²−2x−8; verify relationship.

Factorise

(x−4)(x+2) → zeros 4, −2

✓Sum: 4+(−2)=2=−b/a=2 ✓ Product: 4×(−2)=−8=c/a ✓

More Practice — Solutions

Q11

Max zeros of a quadratic.

✓A quadratic polynomial can have at most 2 zeros, since its graph (a parabola) can cross the x-axis at most twice.

Q12

Zeros of x²−15.

x²=15 → x=±√15

sum=0=−b/a (b=0) ✓ product=−15=c/a ✓

✓x = √15 or x = −√15

Q13

Zeros 2,−3 of x²+(a+1)x+b. Find a,b.

Sum=2+(−3)=−1=−(a+1)

a+1=1 → a=0

Product=2×(−3)=−6=b

b=−6

✓a = 0, b = −6

Q14

Zeros 2+√3, 2−√3.

Sum=4, Product=(2)²−(√3)²=4−3=1

x²−(sum)x+(product)

✓x² − 4x + 1

Q15

1/α+1/β for x²−5x+6.

1/α+1/β=(α+β)/(αβ)

sum=5, product=6

✓5/6

Q16

Zeros of 2x²+3x−2.

Split the middle term

2x²+4x−x−2=2x(x+2)−1(x+2)=(x+2)(2x−1)

✓x = −2 or x = 1/2

Q17

3x²−8x+(2k+1); one zero=7×other. Find k.

Let zeros be α,7α. Sum=8α=8/3 → α=1/3

Product=7α²=(2k+1)/3

Substitute α=1/3

7/9=(2k+1)/3 → 2k+1=7/3 → 2k=4/3

✓k = 2/3

Q18

Zeros of 4x²−4x+1.

Factorise

(2x−1)²=0

✓x = 1/2 (a repeated/equal root)

Q19

kx²+3x+2k: sum=product. Find k.

Sum=−3/k, Product=2k/k=2. Set equal

−3/k=2

✓k = −3/2

Q20

Reciprocal-zero polynomial of 2x²−3x+1.

Original: sum=3/2, product=1/2. New sum=(α+β)/(αβ), new product=1/(αβ)

new sum=(3/2)/(1/2)=3, new product=1/(1/2)=2

✓x² − 3x + 2

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