Questions: Let f(x) be a cubic polynomial with zeros -4, 2, and 1. If the graph of f passes through the point (0,4), write the complete factored form of f(x).

Transcript text: Let $f(x)$ be a cubic polynomial with zeros $-4,2$, and 1. If the graph of $f$ passes through the point $(0,4)$, write th complete factored form of $f(x)$.

Solution

Solution Steps

To find the complete factored form of the cubic polynomial $ f(x) $, we start by using the given zeros $-4$, $2$, and $1$. The polynomial can be expressed as $ f(x) = a(x + 4)(x - 2)(x - 1) $, where $ a $ is a constant. To find the value of $ a $, we use the fact that the graph passes through the point $(0, 4)$. By substituting $ x = 0 $ and $ f(x) = 4 $ into the polynomial, we can solve for $ a $.

Step 1: Define the Polynomial

Given the zeros of the cubic polynomial $ f(x) $ at $ -4, 2, $ and $ 1 $, we can express the polynomial in factored form as: \[ f(x) = a(x + 4)(x - 2)(x - 1) \]

Step 2: Use the Point to Find $ a $

Since the polynomial passes through the point $ (0, 4) $, we substitute $ x = 0 $ and $ f(0) = 4 $: \[ f(0) = a(0 + 4)(0 - 2)(0 - 1) = 4 \] This simplifies to: \[ f(0) = a(4)(-2)(-1) = 8a \] Setting this equal to 4 gives: \[ 8a = 4 \implies a = \frac{1}{2} \]

Step 3: Write the Complete Factored Form

Substituting $ a $ back into the polynomial, we have: \[ f(x) = \frac{1}{2}(x + 4)(x - 2)(x - 1) \]