Question

Pamela factorized the following polynomial:

2x³ + 3x² – 3x – 2

She found the result as (x + 2) (x – 1) (x – 2). Using remainder and factor theorem, verify whether her result is correct. If incorrect, give the correct result.

Answer 1

Given: Pamela’s factorization claim: 2x³ + 3x² – 3x – 2 = (x + 2)(x – 1)(x – 2).

Step-wise calculation:

1. Define f(x) = 2x³ + 3x² – 3x – 2.

2. Use Factor / Remainder Theorem: if (x – a) is a factor then f(a) = 0.

Test x + 2 (root x = –2):

f(–2) = 2(–8) + 3(4) – 3(–2) – 2

= –16 + 12 + 6 – 2

= 0

So, (x + 2) is a factor.

Test x – 1 (root x = 1):

f(1) = 2 + 3 – 3 – 2

= 0

So, (x – 1) is a factor.

Test x – 2 (root x = 2):

f(2) = 2(8) + 3(4) – 3(2) – 2

= 16 + 12 – 6 – 2

= 20

Since f(2) ≠ 0, (x – 2) is NOT a factor (remainder 20).

3. Since (x + 2) and (x – 1) are factors, divide f(x) by (x + 2)(x – 1) = x² + x – 2:

Long division / Synthetic: (2x³ + 3x² – 3x – 2) ÷ (x² + x – 2) gives quotient 2x + 1 and remainder 0.

So, f(x) = (x² + x – 2)(2x + 1) = (x + 2)(x – 1)(2x + 1).

Pamela’s result is incorrect because (x – 2) is not a factor f(2) = 20.

The correct factorization is 2x³ + 3x² – 3x – 2 = (x + 2)(x – 1)(2x + 1).