Find the zeros of the polynomial f(x) = (x

2 − 9)(x

2 + x − 2).

To find the zeros of the polynomial \(f(x) = (x^2 - 9)(x^2 + x - 2)\), we need to determine the values of \(x\) that make the expression equal to zero. 

### Step 1: Set the Polynomial Equal to Zero
Set \(f(x) = 0\):

\[
(x^2 - 9)(x^2 + x - 2) = 0
\]

For the product of two expressions to be zero, at least one of the expressions must be zero. Therefore, we solve each factor separately.

### Step 2: Solve Each Factor

#### 1. Solving \(x^2 - 9 = 0\)

\[
x^2 - 9 = 0
\]

Add 9 to both sides:

\[
x^2 = 9
\]

Take the square root of both sides:

\[
x = \pm 3
\]

So, the zeros from this factor are:

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

#### 2. Solving \(x^2 + x - 2 = 0\)

To solve this quadratic equation, we can factor it. We look for two numbers that multiply to \(-2\) (the constant term) and add to \(1\) (the coefficient of \(x\)).

These numbers are \(2\) and \(-1\). So, we can factor the quadratic as:

\[
x^2 + x - 2 = (x + 2)(x - 1)
\]

Set each factor equal to zero:

\[
x + 2 = 0 \quad \text{or} \quad x - 1 = 0
\]

Solve for \(x\):

\[
x = -2 \quad \text{and} \quad x = 1
\]

### Step 3: Conclusion
The zeros of the polynomial \(f(x) = (x^2 - 9)(x^2 + x - 2)\) are:

\[
x = -3, \quad x = 3, \quad x = -2, \quad x = 1
\]

These are the values of \(x\) that make \(f(x) = 0\).

Would you like more details or have any questions?

### Related Questions:
1. How do you solve a quadratic equation by factoring?
2. What are the properties of a quadratic polynomial?
3. How do you solve quadratic equations using the quadratic formula?
4. What is the significance of the zeros of a polynomial?
5. How can you verify if a given value is a zero of a polynomial?

### Tip:
When factoring quadratic equations, always look for pairs of numbers that multiply to the constant term and add to the coefficient of the linear term.

Discover how to find the zeros of the polynomial f(x) = (x^2 - 9)(x^2 + x - 2) using step-by-step methods. This problem involves solving quadratic equations through factoring and applying the zero product property. Suitable for high school students.

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