While factorizing a given polynomial, using remainder & factor

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While factorizing a given polynomial, using remainder & factor theorem, a student finds that (2x + 1) is a factor of 2x³ + 7x² + 2x – 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.

Also, factorize the given polynomial completely

ICSE Specimen Question Paper (SQP) 2026

a) Check if (2x + 1) is a factor:

Step 1: Let’s consider that (2x +1) is a factor of the given polynomial.

Hence (2 x + 1) = 0

∴ x = $\frac{-1}{2}$

Step 2: Let’s check value of P(x) for x = $\frac{-1}{2}$

Given P(x) = 2x³ + 7x² + 2x – 3

∴ P( $\frac{-1}{2}$ ) = 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

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

= ( $\frac{-1}{4}$ ) + ( $\frac{7}{4}$ ) – 1 – 3 = ( $\frac{6}{4}$ ) – 4

= ( $\frac{3}{2}$ ) – 4 = ( $\frac{- 5}{2}$ )

Therefore, If (2x +1) is not a factor of the given polynomial.

(b) Valid reason for the above answer:

According to the Factor Theorem, (2x +1) will be a factor of the given polynomial, if and only if given polynomial’s value is zero for x = $\frac{-1}{2}$ .

Since, in above test, value of P( $\frac{-1}{2}$ ) is not zero,

Therefore (2x +1) is not a factor to the given polynomial.

(c) Complete factors of the given polynomial:

Step 3: We test for if (2x – 1) is factor of P(x) i.e. x = $\frac{1}{2}$

Let’s check (Px) for x = $\frac{1}{2}$

Given P(x) = 2x³ + 7x² + 2x – 3

∴ P( $\frac{1}{2}$ ) = 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

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

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

= 2 – 2 = 0

Since for X = $\frac{1}{2}$ , value of P(x) is zero, hence, (2x – 1) is a factor

Step 4: Next, to factorize 2x³ + 7x² + 2x – 3 completely, we divide 2x³ + 7x² + 2x – 3 by (2x – 1) by using remainder & factor theorem:

$\frac{2x^3 + 7 x^2 + 2 x - 3}{2x -1}$ = x² + 4x +3

Hence, we can write, 2x³ + 7x² + 2x – 3 = (2x – 1) (x² + 4x +3)

= (2x – 1) (x² + 3x + x +3) (by Middle Term splitting)

= (2x – 1) (x(x + 3) + 1(x +3))

= (2x – 1) (x + 3) (x + 1)

Therefore, given polynomial is (2x – 1) (x + 3) (x + 1) when factorized completely.

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