While factorizing a given polynomial, using remainder & factor

Leave a Comment / Class 10th, ICSE, polynomials / By Sapling Academy

Q) While factorizing a given polynomial, a student finds that (2x + 1) is a factor of 2 x³ + 7 x ² + 2 x – 3
(a) Is the student’s solution correct stating that (2x + 1) is a factor of the given polynomial?
(b) Give a valid reason for your answer.
(c) Factorize the given polynomial completely.

ICSE Specimen Question Paper (SQP)2025

Ans:

(a) Check if (2x + 1) is the factor of polynomial or not:

If (2 x + 1) is the factor of the polynomial, then (2 x + 1) = 0

∴ x = $\frac{- 1}{2}$

Now if (2 x + 1) is the factor of the given polynomial, then x = $\frac{- 1}{2}$ should satisfy the polynomial equation.

Given polynomial is f(x) = 2 x³ + 7 x ² + 2 x – 3

∴ $f (\frac{- 1}{2}$ ) = $2 (\frac{- 1}{2})^3 + 7 (\frac{- 1}{2})^2 + 2 \frac{- 1}{2} - 3$

= $2 (\frac{- 1}{8}) + 7 (\frac{- 1}{4}) + 2 \frac{- 1}{2} - 3$

= $(\frac{- 1}{4}) + (\frac{- 7}{4}) - 1 - 3$

= – 2 – 4 = – 6

Clearly, (2 x + 1) is not the factor of the given polynomial.

(b) Reason:

for (2 x +1) to be the factor of the given polynomial, it should satisfy f(x) = 0

When we keep x = $\frac{- 1}{2}$ , we found f( $\frac{- 1}{2}$ ) = – 6

Since f( $\frac{- 1}{2}$ ) $\neq$ 0, therefore (2 x + 1) is not a factor of the given polynomial.

(c) Factors of the polynomial:

Polynomial is: 2 x ³ + 7 x ² + 2 x – 3 = 0

∴ 2 x ³ – x ² + 8 x ² + 2 x – 3 = 0

∴ x ² (2 x – 1) + 8 x ² – 4 x + 6 x – 3 = 0

∴ x ² (2 x – 1) + 4 x (2 x – 1) + 6 x – 3 = 0

∴ x ² (2 x – 1) + 4 x (2 x – 1) + 3 (2 x – 1) = 0

∴ (2x – 1) (x ² + 4 x + 3) = 0

∴ (2 x – 1) (x + 3) (x + 1) = 0

Therefore, factors of the given polynomial are (2 x – 1), (x + 3) and (x + 1).

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