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uestion 3
Write the complete factored form of the polynomial
f(x), given the indicated zero.

f(x)=x^(3)-9x^(2)+27 x-27
3 is a zero.
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uestion $3$ $\newline$ Write the complete factored form of the polynomial $f(x)$ , given the indicated zero. $\newline$ $f(x)=x^{3}-9 x^{2}+27 x-27$ $\newline$ $3$ is a zero. $\newline$ For the toolbar, press $A L T+F 10(P C)$ or $A L T+F N+F 10$ (Mac).

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Conjugate root theorems

Full solution

Q. uestion

3

\newline

Write the complete factored form of the polynomial

f(x)

, given the indicated zero.

\newline

f(x)=x^{3}-9 x^{2}+27 x-27

\newline

3

is a zero.

\newline

For the toolbar, press

A L T+F 10(P C)

or

A L T+F N+F 10

(Mac).

Identify Zero Factor: Since $3$ is a zero, $(x - 3)$ is a factor of $f(x)$ . Perform polynomial division to divide $f(x)$ by $(x - 3)$ .
Perform Polynomial Division: Divide $f(x)$ by $(x - 3)$ using synthetic division or long division.
Synthetic Division Setup: Synthetic division setup: $\newline$ $3$ | $1$ $-9$ $27$ $-27$ $\newline$ |___ $3$ $-18$ $27$ $\newline$ Result: $1$ $-6$ $1$ $0$ $1$ $1$
Find Quotient: The quotient from the division is $x^2 - 6x + 9$ .
Factor Quadratic: Factor the quadratic $x^2 - 6x + 9$ . It's a perfect square trinomial.
Factor Perfect Square Trinomial: The factored form of $x^2 - 6x + 9$ is $(x - 3)(x - 3)$ or $(x - 3)^2$ .
Combine Factors: Combine the factors to get the complete factored form of $f(x)$ : $(x - 3)(x - 3)^2$ .
Simplify Factored Form: Simplify the factored form to get $f(x) = (x - 3)^3$ .

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