Question

19 find the zeros of the polynomial function below.

(p(x) = (x^2 + 4x + 3)(x^2 - 4))

plot all the zeros (x-intercepts) of the polynomial in the interactive graph below

Explanation:

🆕 New Concept Discovered: Finding Zeros of Polynomials
Finding where a polynomial equals zero by factoring

Step 1: Set the polynomial to zero

To find the zeros (or \(x\)-intercepts) of the polynomial function, we set \(p(x) = 0\):

\[
(x^2 + 4x + 3)(x^2 - 4) = 0
\]

This means we can find the zeros by setting each individual factor to zero:

\[
x^2 + 4x + 3 = 0 \quad \text{or} \quad x^2 - 4 = 0
\]

Step 2: Solve the first quadratic factor

Factor the quadratic expression \(x^2 + 4x + 3\). We look for two numbers that multiply to \(3\) and add up to \(4\). These numbers are \(1\) and \(3\):

\[
(x + 1)(x + 3) = 0
\]

Setting each factor to zero gives:

\[
x = -1 \quad \text{and} \quad x = -3
\]

Step 3: Solve the second quadratic factor

Factor the difference of squares \(x^2 - 4\):

\[
(x - 2)(x + 2) = 0
\]

Setting each factor to zero gives:

\[
x = 2 \quad \text{and} \quad x = -2
\]

Step 4: Identify the points to plot

The zeros of the polynomial are \(x = -3, -2, -1, 2\).

To plot these on the graph, place points at the following coordinates on the \(x\)-axis:

• \((-3, 0)\)
• \((-2, 0)\)
• \((-1, 0)\)
• \((2, 0)\)

Answer:

The zeros of the polynomial are:
\[
x = -3, -2, -1, 2
\]

Plot points on the \(x\)-axis at:

• \(-3\)
• \(-2\)
• \(-1\)
• \(2\)