While factorizing a given polynomial, using remainder & factor theorem, a student finds

Question

Q) 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

Sapling Academy · Accepted Answer

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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