Question

Polynomial Functions & Graphs Unit Test

Decompose the polynomial into linear factors (x ^ 2 - 7x + 6)(x ^ 2 + 3x - 18) (1 point)

Answer 1

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

1. **Factor the first polynomial: \( x^2 - 7x + 6 \)**

We need to find two numbers that multiply to \( 6 \) (the constant term) and add up to \( -7 \) (the coefficient of \( x \)). The numbers \( -1 \) and \( -6 \) satisfy these conditions.

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

2. **Factor the second polynomial: \( x^2 + 3x - 18 \)**

Now, we need to find two numbers that multiply to \( -18 \) and add up to \( 3 \). The numbers \( 6 \) and \( -3 \) satisfy these conditions.

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

3. **Combine the factors:**

Now, we can write the entire polynomial as a product of its linear factors:
\[
(x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 1)(x - 6)(x + 6)(x - 3)
\]

So, the complete decomposition into linear factors is:
\[
(x - 1)(x - 6)(x + 6)(x - 3)
\]