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: \(2x - 5\)
-
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: \(x + 5\)
-
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: \(2x + 7\)
-
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: \(x - 7\)
-
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).
1 answer