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Practice problems
Find the zeros of the Folynomial functions

f(x)=x^(3)+8x^(2)+11 x-20

Practice problems $\newline$ Find the zeros of the Folynomial functions $\newline$ $f(x)=x^{3}+8 x^{2}+11 x-20$

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Quadratic equation with complex roots

Full solution

Q. Practice problems

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Find the zeros of the Folynomial functions

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f(x)=x^{3}+8 x^{2}+11 x-20

Identify Obvious Factors: Look for any obvious factors of the polynomial. $\newline$ Try plugging in simple values like $x=1$ , $x=-1$ to see if they are zeros. $\newline$ $f(1)=1^3+8\cdot1^2+11\cdot1-20 = 1+8+11-20 = 0$ , so $x=1$ is a zero.
Use Synthetic Division: Use synthetic division or long division to divide the polynomial by $(x-1)$ . $\newline$ Divide $f(x)$ by $(x-1)$ to find the other factors. $\newline$ $(x-1) | x^3 + 8x^2 + 11x - 20$ $\newline$ | $x^2 + 9x + 20$ $\newline$ The quotient is $x^2 + 9x + 20$ .
Factor Quadratic Equation: Factor the quadratic equation $x^2 + 9x + 20$ . $\newline$ Look for two numbers that multiply to $20$ and add up to $9$ . $\newline$ The numbers are $5$ and $4$ , so the factors are $(x+5)$ and $(x+4)$ .
Solve for Zeros: Set each factor equal to zero and solve for $x$ . $x - 1 = 0$ gives $x = 1$ $x + 5 = 0$ gives $x = -5$ $x + 4 = 0$ gives $x = -4$
List All Zeros: Write down all the zeros of the polynomial. $\newline$ The zeros are $x = 1$ , $x = -5$ , and $x = -4$ .

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