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The polynomial $p(x) = x^3 - 19x - 30$ has a known factor of $(x + 2)$ . Rewrite $p(x)$ as a product of linear factors.

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Factor using a quadratic pattern

Full solution

Q. The polynomial

p(x) = x^3 - 19x - 30

has a known factor of

(x + 2)

. Rewrite

p(x)

as a product of linear factors.

Polynomial Division: First, let's do polynomial division to divide $p(x)$ by $(x+2)$ .
Set Up Division: Set up the division: $(x^3 - 19x - 30) \div (x + 2)$ .
Divide $x^3$ : Divide $x^3$ by $x$ to get $x^2$ . Multiply $(x + 2)$ by $x^2$ to get $x^3 + 2x^2$ . Subtract this from the original polynomial to get $-2x^2 - 19x$ .
Divide $-2x^2$ : Now, divide $-2x^2$ by $x$ to get $-2x$ . Multiply $(x + 2)$ by $-2x$ to get $-2x^2 - 4x$ . Subtract this from $-2x^2 - 19x$ to get $-15x$ .
Divide $-15x$ : Next, divide $-15x$ by $x$ to get $-15$ . Multiply $(x + 2)$ by $-15$ to get $-15x - 30$ . Subtract this from $-15x - 30$ to get $0$ .
Factor Quadratic: So, the result of the division is $x^2 - 2x - 15$ . Now we need to factor this quadratic.
Find Two Numbers: Looking for two numbers that multiply to $-15$ and add up to $-2$ . Those numbers are $-5$ and $3$ .
Factor Quadratic: Factor the quadratic to get $(x - 5)(x + 3)$ .
Linear Factors: Now we have all the linear factors of $p(x)$ : $(x + 2)(x - 5)(x + 3)$ .

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