Question

1. factoring is the opposite of _______

1. what is the greatest common factor of these terms: \\(4xy^2, 10x^2y^2, 8xy^3\\)

• \\(40xy^2\\)
• \\(2xy^2\\)
• \\(2x^2y^3\\)
• \\(2xy\\)

1. factor the polynomial fully: \\(2x^2 + 26x + 60\\). explain your steps and how you found the factor pairs.

Explanation:

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<reasoning>

Define factoring as an operation

Factoring is the process of breaking down an expression into a product of simpler expressions.

Using the Factoring Definition knowledge point
\[
\text{Factoring is the opposite of multiplying (or expanding).}
\]

Find the greatest common factor of the terms

Using the Greatest Common Factor knowledge point
\[

$$\begin{aligned} &\text{Terms: } 4xy^2, \ 10x^2y^2, \ 8xy^3\\ &\text{GCF of coefficients (4, 10, 8): } 2\\ &\text{GCF of } x \text{ terms } (x, x^2, x): x\\ &\text{GCF of } y \text{ terms } (y^2, y^2, y^3): y^2\\ &\text{GCF} = 2xy^2 \end{aligned}$$

\]

Factor out the greatest common factor

Using the Greatest Common Factor and Factoring Trinomials knowledge points
\[

$$\begin{aligned} &2x^2 + 26x + 60\\ &= 2(x^2 + 13x + 30) \end{aligned}$$

\]

Factor the remaining quadratic trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &\text{Find factors of 30 that add up to 13:}\\ &1 \times 30 \implies 1 + 30 = 31\\ &2 \times 15 \implies 2 + 15 = 17\\ &3 \times 10 \implies 3 + 10 = 13 \quad (\text{Correct pair})\\ &x^2 + 13x + 30 = (x + 3)(x + 10)\\ &2x^2 + 26x + 60 = 2(x + 3)(x + 10) \end{aligned}$$

\]
</reasoning>

<answer>

Question 5

Factoring is the opposite of <blank>multiplying</blank> (or <blank>expanding</blank>).

Question 6

<mcq-option>(A) \(40xy^2\)</mcq-option>
<mcq-correct>(B) \(2xy^2\)</mcq-correct>
<mcq-option>(C) \(2x^2y^3\)</mcq-option>
<mcq-option>(D) \(2xy\)</mcq-option>

Question 7

To factor the polynomial \(2x^2 + 26x + 60\) fully:

1. Factor out the GCF: The greatest common factor of the terms is \(2\). Factoring it out gives:

\[2(x^2 + 13x + 30)\]

1. Find the factor pairs: We need two numbers that multiply to \(30\) and add up to \(13\).

• The factors of \(30\) are: \((1, 30)\), \((2, 15)\), \((3, 10)\), and \((5, 6)\).
• The pair that adds up to \(13\) is \(3\) and \(10\) (since \(3 \times 10 = 30\) and \(3 + 10 = 13\)).

1. Write the factored form:

\[2(x + 3)(x + 10)\]
</answer>

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Answer:

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<reasoning>

Define factoring as an operation

Factoring is the process of breaking down an expression into a product of simpler expressions.

Using the Factoring Definition knowledge point
\[
\text{Factoring is the opposite of multiplying (or expanding).}
\]

Find the greatest common factor of the terms

Using the Greatest Common Factor knowledge point
\[

$$\begin{aligned} &\text{Terms: } 4xy^2, \ 10x^2y^2, \ 8xy^3\\ &\text{GCF of coefficients (4, 10, 8): } 2\\ &\text{GCF of } x \text{ terms } (x, x^2, x): x\\ &\text{GCF of } y \text{ terms } (y^2, y^2, y^3): y^2\\ &\text{GCF} = 2xy^2 \end{aligned}$$

\]

Factor out the greatest common factor

Using the Greatest Common Factor and Factoring Trinomials knowledge points
\[

$$\begin{aligned} &2x^2 + 26x + 60\\ &= 2(x^2 + 13x + 30) \end{aligned}$$

\]

Factor the remaining quadratic trinomial

Using the Factoring Trinomials knowledge point
\[

$$\begin{aligned} &\text{Find factors of 30 that add up to 13:}\\ &1 \times 30 \implies 1 + 30 = 31\\ &2 \times 15 \implies 2 + 15 = 17\\ &3 \times 10 \implies 3 + 10 = 13 \quad (\text{Correct pair})\\ &x^2 + 13x + 30 = (x + 3)(x + 10)\\ &2x^2 + 26x + 60 = 2(x + 3)(x + 10) \end{aligned}$$

\]
</reasoning>

<answer>

Question 5

Factoring is the opposite of <blank>multiplying</blank> (or <blank>expanding</blank>).

Question 6

<mcq-option>(A) \(40xy^2\)</mcq-option>
<mcq-correct>(B) \(2xy^2\)</mcq-correct>
<mcq-option>(C) \(2x^2y^3\)</mcq-option>
<mcq-option>(D) \(2xy\)</mcq-option>

Question 7

To factor the polynomial \(2x^2 + 26x + 60\) fully:

1. Factor out the GCF: The greatest common factor of the terms is \(2\). Factoring it out gives:

\[2(x^2 + 13x + 30)\]

1. Find the factor pairs: We need two numbers that multiply to \(30\) and add up to \(13\).

• The factors of \(30\) are: \((1, 30)\), \((2, 15)\), \((3, 10)\), and \((5, 6)\).
• The pair that adds up to \(13\) is \(3\) and \(10\) (since \(3 \times 10 = 30\) and \(3 + 10 = 13\)).

1. Write the factored form:

\[2(x + 3)(x + 10)\]
</answer>

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