Question
Which of the following expressions is a factor of 2x2+9x−35? Use long division to solve.(1 point) Responses x+5 x plus 5 2x+7 2 x plus 7 2x−5 2 x minus 5 x−7
Answers
Answer
Can you double check Brainly website said 2x+7
Answer
Ok thank you for double checking
Answered by
GPT-4o mini
To determine which expression is a factor of \(2x^2 + 9x - 35\), we can use polynomial long division. We'll test each of the provided options as potential factors.
We'll start by testing \(x + 5\):
**Step 1: Perform Long Division of \(2x^2 + 9x - 35\) by \(x + 5\)**
1. Divide \(2x^2\) by \(x\) to get \(2x\).
2. Multiply \(x + 5\) by \(2x\) to get \(2x^2 + 10x\).
3. Subtract \(2x^2 + 10x\) from \(2x^2 + 9x\):
\[
(2x^2 + 9x) - (2x^2 + 10x) = -x
\]
4. Bring down the \(-35\) to get \(-x - 35\).
5. Divide \(-x\) by \(x\) to get \(-1\).
6. Multiply \(x + 5\) by \(-1\) to get \(-x - 5\).
7. Subtract \(-x - 5\) from \(-x - 35\):
\[
(-x - 35) - (-x - 5) = -30
\]
After performing long division, we have a remainder of \(-30\), meaning \(x + 5\) is **not** a factor.
Now, let's test \(2x - 5\):
**Step 2: Perform Long Division of \(2x^2 + 9x - 35\) by \(2x - 5\)**
1. Divide \(2x^2\) by \(2x\) to get \(x\).
2. Multiply \(2x - 5\) by \(x\) to get \(2x^2 - 5x\).
3. Subtract \(2x^2 - 5x\) from \(2x^2 + 9x\):
\[
(2x^2 + 9x) - (2x^2 - 5x) = 14x
\]
4. Bring down the \(-35\) to get \(14x - 35\).
5. Divide \(14x\) by \(2x\) to get \(7\).
6. Multiply \(2x - 5\) by \(7\) to get \(14x - 35\).
7. Subtract \(14x - 35\) from \(14x - 35\):
\[
(14x - 35) - (14x - 35) = 0
\]
Since we have a remainder of 0, \(2x - 5\) is a factor.
Therefore, the answer is:
**\(2x - 5\)** (the expression that is a factor of \(2x^2 + 9x - 35\)).
We'll start by testing \(x + 5\):
**Step 1: Perform Long Division of \(2x^2 + 9x - 35\) by \(x + 5\)**
1. Divide \(2x^2\) by \(x\) to get \(2x\).
2. Multiply \(x + 5\) by \(2x\) to get \(2x^2 + 10x\).
3. Subtract \(2x^2 + 10x\) from \(2x^2 + 9x\):
\[
(2x^2 + 9x) - (2x^2 + 10x) = -x
\]
4. Bring down the \(-35\) to get \(-x - 35\).
5. Divide \(-x\) by \(x\) to get \(-1\).
6. Multiply \(x + 5\) by \(-1\) to get \(-x - 5\).
7. Subtract \(-x - 5\) from \(-x - 35\):
\[
(-x - 35) - (-x - 5) = -30
\]
After performing long division, we have a remainder of \(-30\), meaning \(x + 5\) is **not** a factor.
Now, let's test \(2x - 5\):
**Step 2: Perform Long Division of \(2x^2 + 9x - 35\) by \(2x - 5\)**
1. Divide \(2x^2\) by \(2x\) to get \(x\).
2. Multiply \(2x - 5\) by \(x\) to get \(2x^2 - 5x\).
3. Subtract \(2x^2 - 5x\) from \(2x^2 + 9x\):
\[
(2x^2 + 9x) - (2x^2 - 5x) = 14x
\]
4. Bring down the \(-35\) to get \(14x - 35\).
5. Divide \(14x\) by \(2x\) to get \(7\).
6. Multiply \(2x - 5\) by \(7\) to get \(14x - 35\).
7. Subtract \(14x - 35\) from \(14x - 35\):
\[
(14x - 35) - (14x - 35) = 0
\]
Since we have a remainder of 0, \(2x - 5\) is a factor.
Therefore, the answer is:
**\(2x - 5\)** (the expression that is a factor of \(2x^2 + 9x - 35\)).
Answered by
GPT-4o mini
Let's thoroughly check the polynomial \(2x^2 + 9x - 35\) to confirm which expression is a factor. We will factor it completely and then see if \(2x + 7\) is indeed a factor.
### Factoring the Quadratic Expression
We can factor \(2x^2 + 9x - 35\) by using the method of grouping.
1. **Identify the product and sum:**
- We need two numbers that multiply to \(2 \times -35 = -70\) (the product) and add up to \(9\) (the sum).
- The numbers that fulfill this are \(14\) and \(-5\) since \(14 \times -5 = -70\) and \(14 + (-5) = 9\).
2. **Rewriting the quadratic:**
So, we can rewrite \(9x\) using \(14x\) and \(-5x\):
\[
2x^2 + 14x - 5x - 35
\]
3. **Grouping:**
Now, group the terms:
\[
(2x^2 + 14x) + (-5x - 35)
\]
4. **Factor out common terms:**
- From the first group \(2x(x + 7)\) and from the second group \(-5(x + 7)\):
\[
2x(x + 7) - 5(x + 7)
\]
5. **Combine the groups:**
Factor out the common binomial factor:
\[
(2x - 5)(x + 7)
\]
### Conclusion:
The fully factored form of the polynomial \(2x^2 + 9x - 35\) is:
\[
(2x - 5)(x + 7)
\]
From this factorization, we can see that:
- \(2x - 5\) is a factor.
- \(x + 7\) is also a factor.
Therefore, the previously suggested factor \(2x + 7\) cannot be correct as we find \(x + 7\) but not \(2x + 7\) is a factor of the original polynomial.
The factors are **\(2x - 5\)** and **\(x + 7\)**. The option \(2x + 7\) does not match either of those factors.
So, the correct factor is indeed \(2x - 5\), not \(2x + 7\).
### Factoring the Quadratic Expression
We can factor \(2x^2 + 9x - 35\) by using the method of grouping.
1. **Identify the product and sum:**
- We need two numbers that multiply to \(2 \times -35 = -70\) (the product) and add up to \(9\) (the sum).
- The numbers that fulfill this are \(14\) and \(-5\) since \(14 \times -5 = -70\) and \(14 + (-5) = 9\).
2. **Rewriting the quadratic:**
So, we can rewrite \(9x\) using \(14x\) and \(-5x\):
\[
2x^2 + 14x - 5x - 35
\]
3. **Grouping:**
Now, group the terms:
\[
(2x^2 + 14x) + (-5x - 35)
\]
4. **Factor out common terms:**
- From the first group \(2x(x + 7)\) and from the second group \(-5(x + 7)\):
\[
2x(x + 7) - 5(x + 7)
\]
5. **Combine the groups:**
Factor out the common binomial factor:
\[
(2x - 5)(x + 7)
\]
### Conclusion:
The fully factored form of the polynomial \(2x^2 + 9x - 35\) is:
\[
(2x - 5)(x + 7)
\]
From this factorization, we can see that:
- \(2x - 5\) is a factor.
- \(x + 7\) is also a factor.
Therefore, the previously suggested factor \(2x + 7\) cannot be correct as we find \(x + 7\) but not \(2x + 7\) is a factor of the original polynomial.
The factors are **\(2x - 5\)** and **\(x + 7\)**. The option \(2x + 7\) does not match either of those factors.
So, the correct factor is indeed \(2x - 5\), not \(2x + 7\).
Answered by
GPT-4o mini
You're welcome! I'm glad I could help clarify that for you. If you have any more questions about polynomials or any other topic, feel free to ask!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.