Question

Which of the following expressions is a factor of 2x^2+9x−35? Use long division to solve.

Answer 1

To factor the expression \(2x^2 + 9x - 35\), we can look for factors using polynomial long division or the factorization method. However, first, let’s see if we can find its factors directly.

We need to find two numbers that multiply to \(2 \cdot (-35) = -70\) and add to \(9\). The numbers that fulfill these requirements are \(14\) and \(-5\).

This allows us to rewrite \(9x\) as \(14x - 5x\):

\[
2x^2 + 14x - 5x - 35
\]

Next, we group the terms:

\[
(2x^2 + 14x) + (-5x - 35)
\]

Now, factor each group:

\[
2x(x + 7) - 5(x + 7)
\]

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

\[
(2x - 5)(x + 7)
\]

Thus, the expression \(2x^2 + 9x - 35\) can be factored as:

\[
(2x - 5)(x + 7)
\]

To verify this factorization, we can expand \((2x - 5)(x + 7)\):

\[
2x \cdot x + 2x \cdot 7 - 5 \cdot x - 5 \cdot 7 = 2x^2 + 14x - 5x - 35 = 2x^2 + 9x - 35
\]

### Conclusion

Thus, the factors of \(2x^2 + 9x - 35\) are \((2x - 5)\) and \((x + 7)\). The expressions that are factors of \(2x^2 + 9x - 35\) are \(2x - 5\) and \(x + 7\).