Question

Samantha factored the polynomial x 4 + 3 x 3 – 8x – 24 in her Algebra 2 class and found the factors to be (x 3 – 8)(x + 3).

She asked her friend Jeremy to check her work.

Jeremy explained to Samantha that the factor (x 3 – 8) can be factored further as a __
​
of cubes.

Answer 1

Given :- Samantha factored the polynomial x⁴ + 3x³ – 8x – 24 and found the factors to be (x³ – 8)(x + 3).

To Find :- (x³ – 8) can be factored further as a __ ?

Solution :-

dividing the given polynomial x⁴ + 3x³ – 8x – 24 by (x + 3) we get,

x + 3 ) x⁴ + 3x³ – 8x – 24 ( x³ - 8

x⁴ + 3x³

- 8x - 24

- 8x - 24

0

So,

→ Quotient = (x³ - 8)

→ Remainder = 0

then,

→ x⁴ + 3x³ – 8x – 24 = (x + 3)(x³ - 8)

now, factorising quotient (x³ - 8) further,

→ (x³ - 8)

→ x³ - 2 × 2 × 2

→ x³ - 2³

→ (x³ - 2³)

using (a³ - b³) = (a - b)(a² + b² + ab)

→ (x - 2)(x² + 2² + 2x)

→ (x - 2)(x² + 2x + 4)

therefore, factorising the given polynomial we get,

→ x⁴ + 3x³ – 8x – 24

→ (x + 3)(x - 2)(x² + 2x + 4)

hence, (x³ - 8) can be factored further as (x - 2)(x² + 2x + 4) .

Learn more :-

JEE mains Question :-

https://brainly.in/question/22246812

. Find all the zeroes of the polynomial x4

– 5x3 + 2x2+10x-8, if two of its zeroes are 4 and 1.

https://brainly.in/question/39026698