Question
Jodie was working on factoring by GCF. She factored the following polynomial expression: 2x^4−10x^3+12x^2
Jodie's work is shown below:
2x^4−10x^3+12x^2
2x(x^3−5x^2+6x)
A. Describe Jodie's error in factoring out the greatest common monomial factor.
B. Correct the factoring error and factor completely. Remember that you must show all of your work and can upload a picture of your handwritten work.
(5 points)
0 / 10000 Word Limit
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Jodie's work is shown below:
2x^4−10x^3+12x^2
2x(x^3−5x^2+6x)
A. Describe Jodie's error in factoring out the greatest common monomial factor.
B. Correct the factoring error and factor completely. Remember that you must show all of your work and can upload a picture of your handwritten work.
(5 points)
0 / 10000 Word Limit
Skip to navigation
Answers
GPT-4o mini
A. Jodie's error in factoring out the greatest common factor (GCF) lies in the second part of her factoring process. While she correctly identified the GCF of the polynomial expression \(2x^4 - 10x^3 + 12x^2\) as \(2x\), she made a mistake in factoring the remaining expression \(x^3 - 5x^2 + 6x\).
When she factored out \(2x\), she should have divided each term of the original polynomial expression by \(2x\), resulting in the correct remaining expression.
Here’s the correct process step-by-step:
1. **Identify the GCF**: The GCF of \(2x^4\), \(-10x^3\), and \(12x^2\) is \(2x\).
2. **Factor out the GCF**: Divide each term by \(2x\):
- \( \frac{2x^4}{2x} = x^3 \)
- \( \frac{-10x^3}{2x} = -5x^2 \)
- \( \frac{12x^2}{2x} = 6x \)
So, correctly factoring out \(2x\) gives:
\[
2x(x^3 - 5x^2 + 6x)
\]
The next step is to factor the remaining polynomial \(x^3 - 5x^2 + 6x\) completely.
B. To factor \(x^3 - 5x^2 + 6x\) completely, follow these steps:
1. **Factor out common terms**: First, observe that \(x\) is a common factor:
\[
x(x^2 - 5x + 6)
\]
2. **Factor the quadratic**: Now we need to factor the quadratic \(x^2 - 5x + 6\). We look for two numbers that multiply to \(6\) and add up to \(-5\). The numbers \(-2\) and \(-3\) fit:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
3. **Combine all factors**: Now we can combine all parts of the factors:
\[
2x(x^2 - 5x + 6) = 2x(x - 2)(x - 3)
\]
So, the completely factored form of the polynomial \(2x^4 - 10x^3 + 12x^2\) is:
\[
\boxed{2x(x - 2)(x - 3)}
\]
This is the final answer, showing all work unambiguously to correct Jodie's initial error.
When she factored out \(2x\), she should have divided each term of the original polynomial expression by \(2x\), resulting in the correct remaining expression.
Here’s the correct process step-by-step:
1. **Identify the GCF**: The GCF of \(2x^4\), \(-10x^3\), and \(12x^2\) is \(2x\).
2. **Factor out the GCF**: Divide each term by \(2x\):
- \( \frac{2x^4}{2x} = x^3 \)
- \( \frac{-10x^3}{2x} = -5x^2 \)
- \( \frac{12x^2}{2x} = 6x \)
So, correctly factoring out \(2x\) gives:
\[
2x(x^3 - 5x^2 + 6x)
\]
The next step is to factor the remaining polynomial \(x^3 - 5x^2 + 6x\) completely.
B. To factor \(x^3 - 5x^2 + 6x\) completely, follow these steps:
1. **Factor out common terms**: First, observe that \(x\) is a common factor:
\[
x(x^2 - 5x + 6)
\]
2. **Factor the quadratic**: Now we need to factor the quadratic \(x^2 - 5x + 6\). We look for two numbers that multiply to \(6\) and add up to \(-5\). The numbers \(-2\) and \(-3\) fit:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
3. **Combine all factors**: Now we can combine all parts of the factors:
\[
2x(x^2 - 5x + 6) = 2x(x - 2)(x - 3)
\]
So, the completely factored form of the polynomial \(2x^4 - 10x^3 + 12x^2\) is:
\[
\boxed{2x(x - 2)(x - 3)}
\]
This is the final answer, showing all work unambiguously to correct Jodie's initial error.