Question
Identify the factors of the polynomial 4x2 + 14x - 30 when factored completely.
2x + 3
x - 5
x + 5
2x - 3
2
4
2x + 3
x - 5
x + 5
2x - 3
2
4
Answers
GPT-4o mini
To factor the polynomial \(4x^2 + 14x - 30\) completely, we'll start by looking for two numbers that multiply to the product of the leading coefficient (4) and the constant term (-30), which is \(4 \times -30 = -120\), and add to the middle coefficient (14).
1. **Finding Suitable Numbers**: We need to find two numbers that multiply to -120 and add up to 14. The pair of numbers that satisfies these conditions is 20 and -6, since:
\[
20 \times (-6) = -120 \quad \text{and} \quad 20 + (-6) = 14
\]
2. **Rewriting the Middle Term**: We can now rewrite the polynomial:
\[
4x^2 + 20x - 6x - 30
\]
3. **Grouping the Terms**: Next, we group the terms:
\[
(4x^2 + 20x) + (-6x - 30)
\]
4. **Factoring Each Group**:
- From the first group, \(4x^2 + 20x\), we can factor out \(4x\):
\[
4x(x + 5)
\]
- From the second group, \(-6x - 30\), we can factor out \(-6\):
\[
-6(x + 5)
\]
5. **Combining the Factored Groups**: Now, we combine the factored expressions:
\[
4x(x + 5) - 6(x + 5)
\]
Since \((x + 5)\) is a common factor:
\[
(x + 5)(4x - 6)
\]
6. **Factoring Further**: The term \(4x - 6\) can be factored further:
\[
4x - 6 = 2(2x - 3)
\]
Thus, we can rewrite the polynomial as:
\[
4x^2 + 14x - 30 = (x + 5)(2(2x - 3))
\]
So, the completely factored form of the polynomial is:
\[
2(x + 5)(2x - 3)
\]
**Final Factors**: From the complete factorization, the factors of the polynomial \(4x^2 + 14x - 30\) when factored completely are:
- \(2\)
- \(x + 5\)
- \(2x - 3\)
The choices provided in your question do not all match the complete factors of the polynomial, but you can choose:
- \(2\)
- \(x + 5\) (which is not listed, but is a factor)
- \(2x - 3\) (which is in your list).
So, the relevant factors from your list are:
- \(2\)
- \(2x - 3\)
Thus, \(2\) and \(2x - 3\) are part of the complete factorization.
1. **Finding Suitable Numbers**: We need to find two numbers that multiply to -120 and add up to 14. The pair of numbers that satisfies these conditions is 20 and -6, since:
\[
20 \times (-6) = -120 \quad \text{and} \quad 20 + (-6) = 14
\]
2. **Rewriting the Middle Term**: We can now rewrite the polynomial:
\[
4x^2 + 20x - 6x - 30
\]
3. **Grouping the Terms**: Next, we group the terms:
\[
(4x^2 + 20x) + (-6x - 30)
\]
4. **Factoring Each Group**:
- From the first group, \(4x^2 + 20x\), we can factor out \(4x\):
\[
4x(x + 5)
\]
- From the second group, \(-6x - 30\), we can factor out \(-6\):
\[
-6(x + 5)
\]
5. **Combining the Factored Groups**: Now, we combine the factored expressions:
\[
4x(x + 5) - 6(x + 5)
\]
Since \((x + 5)\) is a common factor:
\[
(x + 5)(4x - 6)
\]
6. **Factoring Further**: The term \(4x - 6\) can be factored further:
\[
4x - 6 = 2(2x - 3)
\]
Thus, we can rewrite the polynomial as:
\[
4x^2 + 14x - 30 = (x + 5)(2(2x - 3))
\]
So, the completely factored form of the polynomial is:
\[
2(x + 5)(2x - 3)
\]
**Final Factors**: From the complete factorization, the factors of the polynomial \(4x^2 + 14x - 30\) when factored completely are:
- \(2\)
- \(x + 5\)
- \(2x - 3\)
The choices provided in your question do not all match the complete factors of the polynomial, but you can choose:
- \(2\)
- \(x + 5\) (which is not listed, but is a factor)
- \(2x - 3\) (which is in your list).
So, the relevant factors from your list are:
- \(2\)
- \(2x - 3\)
Thus, \(2\) and \(2x - 3\) are part of the complete factorization.