Question
Jodie was working on factoring by GCF. She factored the following polynomial expression: 2x4−10x3+12x2
Jodie's work is shown below:
2x4−10x3+12x2
2x(x3−5x2+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:
2x4−10x3+12x2
2x(x3−5x2+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
page 4 of 4
Answers
GPT-4o mini
A. Jodie's error in factoring by GCF is that she correctly identified the greatest common factor (GCF) as \(2x\), but she made a mistake in the terms inside the parentheses after factoring it out. The expression \(x^3 - 5x^2 + 6x\) should not have included \(x\) as a factor of the constant term and should reflect the correct division of each term by the GCF.
To clarify:
1. The original polynomial is \(2x^4 - 10x^3 + 12x^2\).
2. The GCF of the terms is indeed \(2x^2\) (not just \(2x\)).
3. Jodie should have factored \(2x^2\), which would result in:
\[
2x^2(x^2 - 5x + 6)
\]
B. To correct the factoring error and factor completely, follow these steps:
1. Identify the GCF of each term in the expression \(2x^4 - 10x^3 + 12x^2\):
- The GCF is \(2x^2\).
2. Factor out \(2x^2\):
\[
2x^4 - 10x^3 + 12x^2 = 2x^2(x^2 - 5x + 6)
\]
3. Now, we need to factor the quadratic expression \(x^2 - 5x + 6\):
- We look for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-2\) and \(-3\) work. Therefore, we can factor this quadratic as:
\[
(x - 2)(x - 3)
\]
4. Putting it all together, the completely factored form of the polynomial is:
\[
2x^2(x - 2)(x - 3)
\]
Thus, the complete and correct factoring of the original polynomial \(2x^4 - 10x^3 + 12x^2\) is:
\[
2x^2(x - 2)(x - 3)
\]
To clarify:
1. The original polynomial is \(2x^4 - 10x^3 + 12x^2\).
2. The GCF of the terms is indeed \(2x^2\) (not just \(2x\)).
3. Jodie should have factored \(2x^2\), which would result in:
\[
2x^2(x^2 - 5x + 6)
\]
B. To correct the factoring error and factor completely, follow these steps:
1. Identify the GCF of each term in the expression \(2x^4 - 10x^3 + 12x^2\):
- The GCF is \(2x^2\).
2. Factor out \(2x^2\):
\[
2x^4 - 10x^3 + 12x^2 = 2x^2(x^2 - 5x + 6)
\]
3. Now, we need to factor the quadratic expression \(x^2 - 5x + 6\):
- We look for two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-2\) and \(-3\) work. Therefore, we can factor this quadratic as:
\[
(x - 2)(x - 3)
\]
4. Putting it all together, the completely factored form of the polynomial is:
\[
2x^2(x - 2)(x - 3)
\]
Thus, the complete and correct factoring of the original polynomial \(2x^4 - 10x^3 + 12x^2\) is:
\[
2x^2(x - 2)(x - 3)
\]