Question

20 multiple choice 1 point given that (x-4) is one of the factors of $f(x)=x^3 - x^2 - 10x - 8$, find all solutions of f(x). \bigcirc \\{4\\} \bigcirc \\{-1, -2\\} \bigcirc \\{-2, -1, -8, 4\\} \bigcirc \\{-2, -1, 4\\}

Explanation:

Step1: Perform Polynomial Division

Since \((x - 4)\) is a factor of \(f(x)=x^{3}-x^{2}-10x - 8\), we divide \(f(x)\) by \((x - 4)\) using polynomial long division or synthetic division. Using synthetic division with root \(4\):

$$
LATEXBLOCK0
$$

So the quotient polynomial is \(x^{2}+3x + 2\).

Step2: Factor the Quotient

Factor \(x^{2}+3x + 2\). We need two numbers that multiply to \(2\) and add to \(3\), which are \(1\) and \(2\). So \(x^{2}+3x + 2=(x + 1)(x + 2)\).

Step3: Find All Roots

The factors of \(f(x)\) are \((x - 4)(x + 1)(x + 2)\). Setting each factor equal to zero:

• \(x - 4 = 0\) gives \(x = 4\)
• \(x + 1 = 0\) gives \(x = -1\)
• \(x + 2 = 0\) gives \(x = -2\)

So the solutions are \(-2, -1, 4\).

Answer:

\(\{-2, -1, 4\}\) (the last option, e.g., if the last option is D. \(\{-2, -1, 4\}\), then D. \(\{-2, -1, 4\}\))