To determine which of the given expressions is a factor of the polynomial [tex]\(2x^2 + 9x - 35\)[/tex], we can use polynomial long division. We will divide the polynomial by each expression and see if any of them result in a zero remainder. If an expression results in a zero remainder, then it is a factor of the polynomial.
### 1. Test [tex]\(2x + 7\)[/tex]
Step 1: Set up the division:
[tex]\[
\begin{array}{r|rr}
2x + 7 & 2x^2 + 9x - 35 \\
& 2x - \frac{1}{7}
\end{array}
\][/tex]
Step 2: Perform the division:
- Divide the leading term of the polynomial [tex]\(2x^2\)[/tex] by the leading term of the divisor [tex]\(2x\)[/tex], which gives [tex]\(x\)[/tex]:
[tex]\[
2x^2 \div (2x) = x
\][/tex]
- Multiply [tex]\(2x + 7\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
(x)(2x + 7) = 2x^2 + 7x
\][/tex]
- Subtract this product from the original polynomial:
[tex]\[
2x^2 + 9x - 35 - (2x^2 + 7x) = 2x^2 + 9x - 2x^2 - 7x - 35 = 2x + 7x - 35 - 7x = 2x - 35
\][/tex]
Step 3: Divide the next term:
[tex]\[
2x \div (2x) = 1
\x = 1
\][/tex]
Step 4: Multiply [tex]\(2x +7 \)[/tex]by [tex]\(2x +7 1 \)[/tex]
[tex]\[
\left(1\right)(2x + 7)=2x +7
\][/tex]
Subtract term
[tex]\[
2x -\left( 2x +7 \right) -35 -2x = 42
\][/tex]
The remainder isn't zero. [tex]\(2x + 7\)[/tex] isn't a factor.
### 2. Test [tex]\(2x - 5\)[/tex]
We follow a similar process as above:
Step 1: Set up the division:
[tex]\[
\begin{array}{r|rr}
2x -5 & 2x^2 + 9x -35 \\
\\
\end{array}
\][/tex]
Step 2: Divide the leading term:
- [tex]\( 2x^2 \div (2x) = x\)[/tex]:
[tex]\[
(x)(2x - 5) = 2x^2 - 5x
\][/tex]
Step 3: Subtract:
[tex]\[
2x^2 + 9x - 35 - (2x^2 - 5x) = 2x^2 + 5x - 35 -2x^2 + 5x = 14 x -35
Divide term
\[
14x \div (2x) = 7
\][/tex]
Step 4: Multiply dividends
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