Question

factor using polynomial division
the polynomial $p(x) = x^3 - 21x - 20$ has a known factor of $(x - 5)$.
rewrite $p(x)$ as a product of linear factors.
$p(x) = \square$

Explanation:

Step1: Divide \( p(x) \) by \( (x - 5) \)

We use polynomial long division to divide \( x^3 - 21x - 20 \) by \( x - 5 \).
The dividend is \( x^3 + 0x^2 - 21x - 20 \) and the divisor is \( x - 5 \).
First term: \( \frac{x^3}{x} = x^2 \). Multiply \( x - 5 \) by \( x^2 \) to get \( x^3 - 5x^2 \). Subtract from the dividend:
\( (x^3 + 0x^2 - 21x - 20) - (x^3 - 5x^2) = 5x^2 - 21x - 20 \).
Second term: \( \frac{5x^2}{x} = 5x \). Multiply \( x - 5 \) by \( 5x \) to get \( 5x^2 - 25x \). Subtract:
\( (5x^2 - 21x - 20) - (5x^2 - 25x) = 4x - 20 \).
Third term: \( \frac{4x}{x} = 4 \). Multiply \( x - 5 \) by \( 4 \) to get \( 4x - 20 \). Subtract:
\( (4x - 20) - (4x - 20) = 0 \).
So, \( \frac{x^3 - 21x - 20}{x - 5} = x^2 + 5x + 4 \).

Step2: Factor the quadratic

Factor \( x^2 + 5x + 4 \). We need two numbers that multiply to 4 and add to 5. Those numbers are 1 and 4.
So, \( x^2 + 5x + 4 = (x + 1)(x + 4) \).

Step3: Write as product of linear factors

Since \( p(x) = (x - 5)(x^2 + 5x + 4) \), substituting the factored quadratic gives \( p(x) = (x - 5)(x + 1)(x + 4) \).

Answer:

\( (x - 5)(x + 1)(x + 4) \)