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$p(x)=x^3+7x^2-36$ has a known factor of $(x+3)$ Rewrite as a product of linear factors.

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Factor using a quadratic pattern

Full solution

Q.

p(x)=x^3+7x^2-36

has a known factor of

(x+3)

Rewrite as a product of linear factors.

Perform Polynomial Division: Since $(x+3)$ is a known factor, perform polynomial division to divide $p(x)$ by $(x+3)$ .
Subtract Multiples: Divide $x^3$ by $x$ to get $x^2$ . Multiply $(x+3)$ by $x^2$ to get $x^3+3x^2$ . Subtract this from the original polynomial to get $4x^2-36$ .
Factor Quadratic Equation: Divide $4x^2$ by $x$ to get $4x$ . Multiply $(x+3)$ by $4x$ to get $4x^2+12x$ . Subtract this from the remaining polynomial to get $-12x-36$ .
Identify Factorization: Divide $-12x$ by $x$ to get $-12$ . Multiply $(x+3)$ by $-12$ to get $-12x-36$ . Subtract this from the remaining polynomial to get $0$ .
Identify Factorization: Divide $-12x$ by $x$ to get $-12$ . Multiply $(x+3)$ by $-12$ to get $-12x-36$ . Subtract this from the remaining polynomial to get $0$ .The result of the division is $x^2+4x-12$ . Factor this quadratic equation.
Identify Factorization: Divide $-12x$ by $x$ to get $-12$ . Multiply $(x+3)$ by $-12$ to get $-12x-36$ . Subtract this from the remaining polynomial to get $0$ .The result of the division is $x^2+4x-12$ . Factor this quadratic equation.Look for two numbers that multiply to $-12$ and add to $4$ . The numbers are $x$ $0$ and $x$ $1$ .
Identify Factorization: Divide $-12x$ by $x$ to get $-12$ . Multiply $(x+3)$ by $-12$ to get $-12x-36$ . Subtract this from the remaining polynomial to get $0$ .The result of the division is $x^2+4x-12$ . Factor this quadratic equation.Look for two numbers that multiply to $-12$ and add to $4$ . The numbers are $x$ $0$ and $x$ $1$ .Factor $x^2+4x-12$ into $x$ $3$ .
Identify Factorization: Divide $-12x$ by $x$ to get $-12$ . Multiply $(x+3)$ by $-12$ to get $-12x-36$ . Subtract this from the remaining polynomial to get $0$ .The result of the division is $x^2+4x-12$ . Factor this quadratic equation.Look for two numbers that multiply to $-12$ and add to $4$ . The numbers are $x$ $0$ and $x$ $1$ .Factor $x^2+4x-12$ into $x$ $3$ .Combine the known factor $(x+3)$ with the factored form of the quotient to get the product of linear factors: $x$ $5$ .

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