Question

Which of the following expressions is a factor of 2x^2+9x−35? Use long division to solve.
A. 2x-5
B. X+5
C. 2x+7
D. X-7

Answer 1

To determine whether any of the given expressions are factors of the quadratic expression \(2x^2 + 9x - 35\), we can use polynomial long division to divide \(2x^2 + 9x - 35\) by each of the proposed factors.

Let's divide \(2x^2 + 9x - 35\) by each option.

### Option A: \(2x - 5\)

1. **Set up the long division:**

\[
\begin{array}{r|l}
& 2x + 10 \\
2x - 5 & 2x^2 + 9x - 35 \\
\hline
& 2x^2 - 5x \\
\hline
& 14x - 35 \\
& 14x - 35 \\
\hline
& 0 \\
\end{array}
\]

The quotient is \(x + 10\) and the remainder is \(0\). Thus, \(2x - 5\) is a factor.

### Option B: \(x + 5\)

1. **Set up the long division:**

\[
\begin{array}{r|l}
& 2x + -1 \\
x + 5 & 2x^2 + 9x - 35 \\
\hline
& 2x^2 + 10x \\
\hline
& -x - 35 \\
& -x - 5 \\
\hline
& -30 \\
\end{array}
\]

The remainder is \(-30\). Thus, \(x + 5\) is not a factor.

### Option C: \(2x + 7\)

1. **Set up the long division:**

\[
\begin{array}{r|l}
& x + 1 \\
2x + 7 & 2x^2 + 9x - 35 \\
\hline
& 2x^2 + 7x \\
\hline
& 2x - 35 \\
& 2x + 7 \\
\hline
& -42 \\
\end{array}
\]

The remainder is \(-42\). Thus, \(2x + 7\) is not a factor.

### Option D: \(x - 7\)

1. **Set up the long division:**

\[
\begin{array}{r|l}
& 2x + 23 \\
x - 7 & 2x^2 + 9x - 35 \\
\hline
& 2x^2 - 14x \\
\hline
& 23x - 35 \\
& 23x - 161 \\
\hline
& 126 \\
\end{array}
\]

The remainder is \(126\). Thus, \(x - 7\) is not a factor.


### Conclusion
The only expression from the options that is a factor of \(2x^2 + 9x - 35\) is \(2x - 5\) (A).