Question
Factor completely 3x2 − 11x + 8.
(x + 4)(3x − 2)
(x − 2)(3x − 4)
(x + 1)(3x − 8)
(x − 1)(3x − 8)
Answers
GPT-4o mini
To factor the quadratic expression \(3x^2 - 11x + 8\), we will look for two numbers that multiply to \(3 \times 8 = 24\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(-11\) (the coefficient of \(x\)).
The two numbers that satisfy these conditions are \(-3\) and \(-8\), since \(-3 \times -8 = 24\) and \(-3 + -8 = -11\).
Next, we can rewrite the middle term \(-11x\) using \(-3x\) and \(-8x\):
\[
3x^2 - 3x - 8x + 8
\]
Now we can group the terms:
\[
(3x^2 - 3x) + (-8x + 8)
\]
Factoring out the common factors in each group:
\[
3x(x - 1) - 8(x - 1)
\]
Now we see that \((x - 1)\) is a common factor:
\[
(3x - 8)(x - 1)
\]
Thus, the factored form of \(3x^2 - 11x + 8\) is:
\[
(3x - 8)(x - 1)
\]
Since none of the provided answer choices match this factorization, let's check them to see if any match through simplification:
1. \( (x + 4)(3x - 2) \) expands to \(3x^2 + 12x - 2x - 8 = 3x^2 + 10x - 8\) (not a match).
2. \( (x - 2)(3x - 4) \) expands to \(3x^2 - 4x - 6x + 8 = 3x^2 - 10x + 8\) (not a match).
3. \( (x + 1)(3x - 8) \) expands to \(3x^2 - 8x + 3x - 8 = 3x^2 - 5x - 8\) (not a match).
4. \( (x - 1)(3x - 8) \) expands to \(3x^2 - 8x - 3x + 8 = 3x^2 - 11x + 8\) (this is the correct match).
Therefore, the correct factorization is:
\[
(x - 1)(3x - 8)
\]
The answer to your question about the factorization of \(3x^2 - 11x + 8\) is:
\((x - 1)(3x - 8)\).
The two numbers that satisfy these conditions are \(-3\) and \(-8\), since \(-3 \times -8 = 24\) and \(-3 + -8 = -11\).
Next, we can rewrite the middle term \(-11x\) using \(-3x\) and \(-8x\):
\[
3x^2 - 3x - 8x + 8
\]
Now we can group the terms:
\[
(3x^2 - 3x) + (-8x + 8)
\]
Factoring out the common factors in each group:
\[
3x(x - 1) - 8(x - 1)
\]
Now we see that \((x - 1)\) is a common factor:
\[
(3x - 8)(x - 1)
\]
Thus, the factored form of \(3x^2 - 11x + 8\) is:
\[
(3x - 8)(x - 1)
\]
Since none of the provided answer choices match this factorization, let's check them to see if any match through simplification:
1. \( (x + 4)(3x - 2) \) expands to \(3x^2 + 12x - 2x - 8 = 3x^2 + 10x - 8\) (not a match).
2. \( (x - 2)(3x - 4) \) expands to \(3x^2 - 4x - 6x + 8 = 3x^2 - 10x + 8\) (not a match).
3. \( (x + 1)(3x - 8) \) expands to \(3x^2 - 8x + 3x - 8 = 3x^2 - 5x - 8\) (not a match).
4. \( (x - 1)(3x - 8) \) expands to \(3x^2 - 8x - 3x + 8 = 3x^2 - 11x + 8\) (this is the correct match).
Therefore, the correct factorization is:
\[
(x - 1)(3x - 8)
\]
The answer to your question about the factorization of \(3x^2 - 11x + 8\) is:
\((x - 1)(3x - 8)\).