Polynomial Functions & Graphs Unit Test

To decompose the polynomial \((x^2 - 7x + 6)(x^2 + 3x - 18)\) into linear factors, we will first factor each quadratic polynomial individually.

**Step 1: Factor \(x^2 - 7x + 6\)**

We need to find two numbers that multiply to \(6\) (the constant term) and add up to \(-7\) (the coefficient of \(x\)). The numbers that fit this criteria are \(-6\) and \(-1\).

Thus, we can factor:
\[
x^2 - 7x + 6 = (x - 6)(x - 1)
\]

**Step 2: Factor \(x^2 + 3x - 18\)**

Next, we need to find two numbers that multiply to \(-18\) (the constant term) and add up to \(3\) (the coefficient of \(x\)). The numbers that fit this criteria are \(6\) and \(-3\).

Thus, we can factor:
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]

**Step 3: Combine the factors**

Now, we can combine both factorizations together:
\[
(x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 6)(x - 1)(x + 6)(x - 3)
\]

So, the polynomial \((x^2 - 7x + 6)(x^2 + 3x - 18)\) decomposes into linear factors:
\[
\boxed{(x - 6)(x - 1)(x + 6)(x - 3)}
\]