Question

You are asked to use the grouping method to factor x^2 -13+22
.

How should the term -13x
be rewritten?

Answer 1

To factor the expression \( x^2 - 13x + 22 \) using the grouping method, we first need to rewrite the middle term, \(-13x\), as a sum of two terms whose coefficients add up to -13 and whose product equals the constant term (22).

We are looking for two numbers that multiply to \( +22 \) (the constant term) and add up to \( -13 \) (the coefficient of the middle term).

The pairs of factors of 22 are:
- \( 1 \) and \( 22 \)
- \( 2 \) and \( 11 \)

Among these, the pair \( -2 \) and \( -11 \) works because:
- \( -2 + (-11) = -13 \)
- \( -2 \times -11 = 22 \)

Thus, we can rewrite \(-13x\) as \(-2x - 11x\). Now the expression can be rewritten as:

\[
x^2 - 2x - 11x + 22
\]

Now, we can factor by grouping. The rewritten form is:

\[
x^2 - 2x - 11x + 22 = (x^2 - 2x) + (-11x + 22)
\]

Next, we factor out common terms:

\[
= x(x - 2) - 11(x - 2)
\]

Now we can factor out the common binomial:

\[
= (x - 2)(x - 11)
\]

So the factored form of \( x^2 - 13x + 22 \) is:
\[
(x - 2)(x - 11)
\]

In summary, the term \(-13x\) should be rewritten as \(-2x - 11x\).