To solve the problem of dividing the polynomial [tex]\(2x^2 - 3x - 35\)[/tex] by [tex]\(x - 5\)[/tex], follow these steps:
1. Set Up the Problem:
You have the polynomial [tex]\(2x^2 - 3x - 35\)[/tex] to be divided by [tex]\(x - 5\)[/tex].
2. Perform Polynomial Long Division:
- Divide the leading term of the numerator [tex]\(2x^2\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
- Multiply [tex]\(2x\)[/tex] by the entire divisor [tex]\(x - 5\)[/tex]:
[tex]\[
2x \cdot (x - 5) = 2x^2 - 10x
\][/tex]
- Subtract this result from the original numerator:
[tex]\[
(2x^2 - 3x - 35) - (2x^2 - 10x) = 7x - 35
\][/tex]
- Now, divide the new leading term [tex]\(7x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{7x}{x} = 7
\][/tex]
- Multiply [tex]\(7\)[/tex] by the entire divisor [tex]\(x - 5\)[/tex]:
[tex]\[
7 \cdot (x - 5) = 7x - 35
\][/tex]
- Subtract this result from the new numerator [tex]\(7x - 35\)[/tex]:
[tex]\[
(7x - 35) - (7x - 35) = 0
\][/tex]
3. Combine the Quotients:
The quotients from the divisions are [tex]\(2x\)[/tex] and [tex]\(7\)[/tex]. Combining these, we have:
[tex]\[
2x + 7
\][/tex]
So, the final quotient when [tex]\(2x^2 - 3x - 35\)[/tex] is divided by [tex]\(x - 5\)[/tex] is [tex]\(2x + 7\)[/tex].
Therefore, the correct answer is:
A) [tex]\(2x + 7\)[/tex]
