Question
Identify all possible factorizations of the binomial.
8x^2+12x
Responses
6x(x+2)
6 x open paren x plus 2 close paren
3x(2x+4)
3 x open paren 2 x plus 4 close paren
2x2(4x+6)
2 x squared open paren 4 x plus 6 close paren
4(2x+3)
4 times open paren 2 x plus 3 close paren
4x(2x+3)
4 x open paren 2 x plus 3 close paren
2x(4x+6)
8x^2+12x
Responses
6x(x+2)
6 x open paren x plus 2 close paren
3x(2x+4)
3 x open paren 2 x plus 4 close paren
2x2(4x+6)
2 x squared open paren 4 x plus 6 close paren
4(2x+3)
4 times open paren 2 x plus 3 close paren
4x(2x+3)
4 x open paren 2 x plus 3 close paren
2x(4x+6)
Answers
GPT-4o mini
To factor the expression \(8x^2 + 12x\), we can start by identifying the greatest common factor (GCF) of the terms in the binomial.
1. **Find the GCF**:
The coefficients are 8 and 12. The GCF of 8 and 12 is 4. Additionally, both terms contain \(x\). The GCF of the entire expression \(8x^2 + 12x\) is \(4x\).
2. **Factor out the GCF**:
We can factor \(4x\) out of each term:
\[
8x^2 + 12x = 4x(2x + 3)
\]
From this factorization, we can identify the following possible factorizations for \(8x^2 + 12x\):
1. **Correct Factorization**:
- \(4x(2x + 3)\)
We can express \(8x^2 + 12x\) in terms of its factors directly, and we can check the other options given:
- \(6x(x + 2)\) does not reproduce the original expression when expanded.
- \(3x(2x + 4)\) does not reproduce the original expression.
- \(2x^2(4x + 6)\) is not appropriate since \(2x^2\) cannot be factored out correctly.
- \(4(2x + 3)\) is missing the \(x\) factor.
- \(4x(2x + 3)\) is valid and matches our derived factorization.
- \(2x(4x + 6)\) is also incorrect for the same reason previously stated.
Thus, the only correct factorization for \(8x^2 + 12x\) is:
\[
\boxed{4x(2x + 3)}
\]
1. **Find the GCF**:
The coefficients are 8 and 12. The GCF of 8 and 12 is 4. Additionally, both terms contain \(x\). The GCF of the entire expression \(8x^2 + 12x\) is \(4x\).
2. **Factor out the GCF**:
We can factor \(4x\) out of each term:
\[
8x^2 + 12x = 4x(2x + 3)
\]
From this factorization, we can identify the following possible factorizations for \(8x^2 + 12x\):
1. **Correct Factorization**:
- \(4x(2x + 3)\)
We can express \(8x^2 + 12x\) in terms of its factors directly, and we can check the other options given:
- \(6x(x + 2)\) does not reproduce the original expression when expanded.
- \(3x(2x + 4)\) does not reproduce the original expression.
- \(2x^2(4x + 6)\) is not appropriate since \(2x^2\) cannot be factored out correctly.
- \(4(2x + 3)\) is missing the \(x\) factor.
- \(4x(2x + 3)\) is valid and matches our derived factorization.
- \(2x(4x + 6)\) is also incorrect for the same reason previously stated.
Thus, the only correct factorization for \(8x^2 + 12x\) is:
\[
\boxed{4x(2x + 3)}
\]