Question

Which expression is equivalent to the trinomial?
3x^2 + 22x + 7
type your answers to each problem in the table below. The order does not matter.

Answer 1

To factor the trinomial \(3x^2 + 22x + 7\), we look for two numbers that multiply to \(3 \times 7 = 21\) (the product of the leading coefficient and the constant) and add up to \(22\) (the coefficient of the linear term).

The numbers \(21\) and \(1\) fit these conditions since:

- \(21 \cdot 1 = 21\)
- \(21 + 1 = 22\)

Now, we rewrite the trinomial using these numbers to break apart the middle term:

\[
3x^2 + 21x + 1x + 7
\]

Next, we group the terms:

\[
(3x^2 + 21x) + (1x + 7)
\]

Now, we factor out the common coefficients from each group:

\[
3x(x + 7) + 1(x + 7)
\]

Now, we can factor out the common factor \((x + 7)\):

\[
(3x + 1)(x + 7)
\]

Thus, the expression \(3x^2 + 22x + 7\) can be written in factored form as:

\[
(3x + 1)(x + 7)
\]

This is the equivalent expression to the original trinomial.