Question

section 05.3: problem 13 (3 points) results for this submission the answer is not correct. find the equation for a polynomial ( f(x) ) that satisfies the following. - degree 3 - zero at ( x=-1 ) - zero at ( x = - 2 ) - zero at ( x = 2 ) - ( y )-intercept of ( (0,8) ) ( f(x)=-2x^{3}-8x^{2}+2x + 8 ) help (formulas) note: you are in the reduced scoring period. all work counts for 70% of the original. preview my answers submit answers show me another your score was recorded. scores are sent to d2l brightspace every 24 hours. you have attempted this problem 3 times. you received a score of 0% for this attempt. your overall recorded score is 0%. you have unlimited attempts remaining.

Explanation:

Step1: Write the polynomial in factored form

Since the polynomial \(f(x)\) has zeros at \(x = - 1\), \(x=-2\) and \(x = 2\), it can be written as \(f(x)=a(x + 1)(x + 2)(x - 2)\), where \(a\) is a non - zero constant.

Step2: Expand the factored form

First, expand \((x + 2)(x - 2)\) using the difference of squares formula \((m+n)(m - n)=m^{2}-n^{2}\), so \((x + 2)(x - 2)=x^{2}-4\). Then \(f(x)=a(x + 1)(x^{2}-4)=a(x^{3}-4x+x^{2}-4)=a(x^{3}+x^{2}-4x - 4)\).

Step3: Use the y - intercept to find the value of \(a\)

The y - intercept is the value of the function when \(x = 0\). We know that \(f(0)=8\). Substitute \(x = 0\) into \(f(x)=a(x^{3}+x^{2}-4x - 4)\), we get \(f(0)=a(0 + 0-0 - 4)=-4a\). Since \(f(0)=8\), then \(-4a=8\), so \(a=-2\).

Step4: Write the final polynomial

Substitute \(a=-2\) into \(f(x)=a(x^{3}+x^{2}-4x - 4)\), we have \(f(x)=-2(x^{3}+x^{2}-4x - 4)=-2x^{3}-2x^{2}+8x + 8\).

Answer:

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