Question

question 17
find the zeros and give the multiplicity of each. list the zeros in order from least to greatest.
$f(x) = 3x^3 - 6x^2 + 3x$
zero at \boxed{} with multiplicity \boxed{}
zero at \boxed{} with multiplicity \boxed{}

question 18
use a calculator to find the real solution(s). answer exactly.
$x^3 - 2x^2 - 16x + 32 = 0$
$x = \boxed{}$ \boxed{} \boxed{}

question 19
factor the following problem completely. first factor out the greatest common factor, and then factor the remaining trinomial.
$x^4 + 10x^3 + 24x^2$

Explanation:

Question 17

Step1: Factor out GCF

$f(x) = 3x(x^2 - 2x + 1)$

Step2: Factor quadratic

$f(x) = 3x(x-1)^2$

Step3: Find zeros

Set $3x=0$ → $x=0$; set $(x-1)^2=0$ → $x=1$

Step4: Identify multiplicities

$x=0$: exponent 1; $x=1$: exponent 2

Step1: Factor by grouping

$x^2(x-2) -16(x-2)=0$

Step2: Factor out common binomial

$(x^2-16)(x-2)=0$

Step3: Factor difference of squares

$(x-4)(x+4)(x-2)=0$

Step4: Solve for $x$

Set each factor to 0: $x-4=0$; $x+4=0$; $x-2=0$

Step1: Factor out GCF

$x^2(x^2 + 10x + 24)$

Step2: Factor the trinomial

$x^2(x+4)(x+6)$

Answer:

Zero at $\boldsymbol{0}$ with multiplicity $\boldsymbol{1}$
Zero at $\boldsymbol{1}$ with multiplicity $\boldsymbol{2}$

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Question 18