Question

Factorize the cubic polynomial (i) X^3+5x^2–2x-24

Answer 1

Answer:

☆ $\boxed{\sf (x - 2) (x + 3) (x + 4) }$

Step-by-step explanation:

Let the given polynomial be p(x) = x³ + 15x² - 2x - 24

※ Let p(x) = 1

= p(1) = (1)³ + 5(1)² - 2(1) - 24

= p(1) = -20

= p(1) ≠ 0

※ Let p(x) = 2

= p(2) = (2)³ + 5(2)² - 2(2) - 24

= p(2) = 0

= p(2) = 0

→ We know that, if we take g(x) = x - 2 = 0, then it will equal to 2. So, we got our first term and our divisor as (x - 2). We will divide the given polynomial by (x - 2).

→ [Refer to the attachment].

→ We got :- $\bf{ x^2 + 7x + 12}$

→ We will factorise it now :-

= x² + 7x + 12

= x² + 4x + 3x + 12

= x(x + 4) + 3(x + 4)

= (x + 3) (x + 4)

※ At last, we got next two terms.

Final Answer:

→ Factorised form of the given polynomial is (x -2) (x + 3) (x + 4).

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Answer 2

$\huge\mathcal{Solution}$

☆ $\boxed{\sf (x - 2) (x + 3) (x + 4) }$

Let the given polynomial be p(x) = x³ + 15x² - 2x - 24

※ Let p(x) = 1

⟹ p(1) = (1)³ + 5(1)² - 2(1) - 24

⟹ p(1) = -20

⟹ p(1) ≠ 0

※ Let p(x) = 2

⟹ p(2) = (2)³ + 5(2)² - 2(2) - 24

⟹ p(2) = 0

⟹ p(2) = 0

↪ We know that, if we take g(x) = x - 2 = 0, then it will equal to 2. So, we got our first term and our divisor as (x - 2). We will divide the given polynomial by (x - 2).

↪ We got :- $\bf{ x^2 + 7x + 12}$

↪ We will factorise it now :-

⟹ x² + 7x + 12

⟹ x² + 4x + 3x + 12

⟹ x(x + 4) + 3(x + 4)

⟹ (x + 3) (x + 4)

※ At last, we got next two terms.

Final Answer:

↪ Factorised form of the given polynomial is (x -2) (x + 3) (x + 4).

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