Question

1. consider ( f(x)=x(x - 2) ). dilate ( f ) by ( (x - 1) ) to create a new function, ( g ) of higher degree. write ( g ) in factored and standard form.

hint: \dilate\ means \multiply\.
fact. form :
std. form :

1. consider ( f(x)=(x + 3)^{2} ). dilate ( f ) by ( x ) to create a new function, ( g ) of higher degree. write ( g ) in standard form. find the zeros of the function ( g ).

hint: \dilate\ means \multiply\.
( g(x)= )

Explanation:

Step1: Find the factored form of \(g(x)\) for problem 8

Given \(f(x)=x(x - 2)\), and we dilate \(f(x)\) by \((x - 1)\).
By the definition of dilation (multiplication), \(g(x)=x(x - 2)(x - 1)\).

Step2: Expand \(g(x)\) for problem 8

First, multiply \((x-2)(x - 1)\):

$$
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$$

Then multiply by \(x\): \(g(x)=x(x^{2}-3x + 2)=x^{3}-3x^{2}+2x\)

Step3: Find \(g(x)\) for problem 10

Given \(f(x)=(x + 3)^{2}=x^{2}+6x + 9\), and we dilate \(f(x)\) by \(x\).
By the definition of dilation (multiplication), \(g(x)=x(x^{2}+6x + 9)\)

Step4: Expand \(g(x)\) for problem 10

$$
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$$

Step5: Find the zeros of \(g(x)\) for problem 10

Set \(g(x)=0\), so \(x^{3}+6x^{2}+9x = 0\).
Factor out \(x\): \(x(x^{2}+6x + 9)=0\).
Factor the quadratic: \(x(x + 3)^{2}=0\).
Using the zero - product property \(x=0\) or \(x=-3\)

Answer:

• Problem 8:
• Factored form: \(g(x)=x(x - 2)(x - 1)\)
• Standard form: \(g(x)=x^{3}-3x^{2}+2x\)
• Problem 10:
• Standard form: \(g(x)=x^{3}+6x^{2}+9x\)
• Zeros: \(x = 0\) and \(x=-3\)