Question

polynomial description/table to standard
score: 0/3 penalty: 1 off
question
the function $f(x)$ is a cubic function and the zeros of $f(x)$ are $-3$, $-1$ and $2$. assume the leading coefficient of $f(x)$ is $1$. write the equation of the cubic polynomial in standard form.
answer attempt 1 out of 2
$f(x) = $

Explanation:

Step1: Write factored form from roots

If the zeros of a polynomial are \( r_1, r_2, r_3 \), then the factored form is \( f(x)=a(x - r_1)(x - r_2)(x - r_3) \). Here, zeros are \(-3, -1, 2\) and \( a = 1 \), so \( f(x)=(x + 3)(x + 1)(x - 2) \).

Step2: Expand the first two factors

Multiply \((x + 3)(x + 1)\): \( (x + 3)(x + 1)=x^2+x+3x + 3=x^2+4x + 3 \). Now \( f(x)=(x^2+4x + 3)(x - 2) \).

Step3: Expand the product

Multiply \( (x^2+4x + 3)(x - 2) \): \( x^2(x - 2)+4x(x - 2)+3(x - 2)=x^3-2x^2+4x^2-8x+3x - 6 \).

Step4: Combine like terms

Combine like terms: \( x^3+( - 2x^2+4x^2)+( - 8x+3x)-6=x^3+2x^2-5x - 6 \).

Answer:

\( f(x)=x^3 + 2x^2 - 5x - 6 \)