Question

question 1
find the greatest common factor: \\(50x^3y^6, 4x^5y^7, 10x^4y^7\\)

• \\(2xy^6\\)
• \\(5x^4y^6\\)
• \\(10x^3y^7\\)
• \\(2x^3y^6\\)

question 2
factor completely: \\(4q^2rt - 2q^3r^2v + 8qr^2s\\)

• \\(2r(2q^2t - q^3rv + 4qrs)\\)
• \\(2(4q^2rt - q^3r^2v + 4qr^2s)\\)
• \\(-2qr(2qt + q^2rv - 4rs)\\)
• \\(2qr(2qt - q^2rv + 4rs)\\)

question 3
factor the following polynomial expression completely: \\(x(x + 3) + 6(x + 3)\\)

• \\((x - 6)(x - 3)\\)
• \\((x - 6)(x + 3)\\)
• \\((x + 6)(x + 3)\\)
• \\((x + 6)(x + 3)^2\\)

question 4
factor the following expression completely by grouping: \\(x^2 + 3x + 6x + 18\\)

• \\((6x + 3)(x + 6)\\)
• \\((x - 6)(x + 3)\\)
• \\((x + 6)(x + 3)\\)
• \\((x + 6)(x + 3)^2\\)

Explanation:

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<reasoning>

Find the greatest common factor of the terms in Question 1

\[

$$\begin{aligned} &\text{Terms: } 50x^3y^6, \quad 4x^4y^7, \quad 10x^3y^7 \\ &\text{GCF of coefficients } (50, 4, 10) = 2 \\ &\text{GCF of } x \text{ terms } (x^3, x^4, x^3) = x^3 \\ &\text{GCF of } y \text{ terms } (y^6, y^7, y^7) = y^6 \\ &\text{GCF} = 2x^3y^6 \end{aligned}$$

\]

Factor the expression in Question 2 completely

\[

$$\begin{aligned} &\text{Expression: } 4q^2rt - 2q^3r^2v + 8qr^2s \\ &\text{GCF of terms: } 2qr \\ &\text{Factored form: } 2qr(2qt - q^2rv + 4rs) \end{aligned}$$

\]

Factor the polynomial in Question 3 completely

\[

$$\begin{aligned} &\text{Expression: } x(x + 3) + 6(x + 3) \\ &\text{Common binomial factor: } (x + 3) \\ &\text{Factored form: } (x + 6)(x + 3) \end{aligned}$$

\]

Factor the expression in Question 4 by grouping

\[

$$\begin{aligned} &\text{Expression: } x^2 + 3x + 6x + 18 \\ &\text{Group terms: } (x^2 + 3x) + (6x + 18) \\ &\text{Factor each group: } x(x + 3) + 6(x + 3) \\ &\text{Factored form: } (x + 6)(x + 3) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-correct>(A) \(2x^3y^6\)</mcq-correct>
<mcq-option>(B) \(5x^4y^6\)</mcq-option>
<mcq-option>(C) \(10x^3y^7\)</mcq-option>
<mcq-option>(D) \(2x^3y^7\)</mcq-option>

Question 2

<mcq-option>(A) \(2r(2q^2t - q^3rv + 4qrs)\)</mcq-option>
<mcq-option>(B) \(2(4q^2rt - q^3r^2v + 4qr^2s)\)</mcq-option>
<mcq-option>(C) \(-2qr(2qt + q^2rv - 4rs)\)</mcq-option>
<mcq-correct>(D) \(2qr(2qt - q^2rv + 4rs)\)</mcq-correct>

Question 3

<mcq-option>(A) \((x - 6)(x - 3)\)</mcq-option>
<mcq-option>(B) \((x - 6)(x + 3)\)</mcq-option>
<mcq-correct>(C) \((x + 6)(x + 3)\)</mcq-correct>
<mcq-option>(D) \((x + 6)(x + 3)^2\)</mcq-option>

Question 4

<mcq-option>(A) \((6x + 3)(x + 6)\)</mcq-option>
<mcq-option>(B) \((x - 6)(z + 3)\)</mcq-option>
<mcq-correct>(C) \((x + 6)(x + 3)\)</mcq-correct>
<mcq-option>(D) \((x + 6)(x + 3)^2\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the greatest common factor of the terms in Question 1

\[

$$\begin{aligned} &\text{Terms: } 50x^3y^6, \quad 4x^4y^7, \quad 10x^3y^7 \\ &\text{GCF of coefficients } (50, 4, 10) = 2 \\ &\text{GCF of } x \text{ terms } (x^3, x^4, x^3) = x^3 \\ &\text{GCF of } y \text{ terms } (y^6, y^7, y^7) = y^6 \\ &\text{GCF} = 2x^3y^6 \end{aligned}$$

\]

Factor the expression in Question 2 completely

\[

$$\begin{aligned} &\text{Expression: } 4q^2rt - 2q^3r^2v + 8qr^2s \\ &\text{GCF of terms: } 2qr \\ &\text{Factored form: } 2qr(2qt - q^2rv + 4rs) \end{aligned}$$

\]

Factor the polynomial in Question 3 completely

\[

$$\begin{aligned} &\text{Expression: } x(x + 3) + 6(x + 3) \\ &\text{Common binomial factor: } (x + 3) \\ &\text{Factored form: } (x + 6)(x + 3) \end{aligned}$$

\]

Factor the expression in Question 4 by grouping

\[

$$\begin{aligned} &\text{Expression: } x^2 + 3x + 6x + 18 \\ &\text{Group terms: } (x^2 + 3x) + (6x + 18) \\ &\text{Factor each group: } x(x + 3) + 6(x + 3) \\ &\text{Factored form: } (x + 6)(x + 3) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-correct>(A) \(2x^3y^6\)</mcq-correct>
<mcq-option>(B) \(5x^4y^6\)</mcq-option>
<mcq-option>(C) \(10x^3y^7\)</mcq-option>
<mcq-option>(D) \(2x^3y^7\)</mcq-option>

Question 2

<mcq-option>(A) \(2r(2q^2t - q^3rv + 4qrs)\)</mcq-option>
<mcq-option>(B) \(2(4q^2rt - q^3r^2v + 4qr^2s)\)</mcq-option>
<mcq-option>(C) \(-2qr(2qt + q^2rv - 4rs)\)</mcq-option>
<mcq-correct>(D) \(2qr(2qt - q^2rv + 4rs)\)</mcq-correct>

Question 3

<mcq-option>(A) \((x - 6)(x - 3)\)</mcq-option>
<mcq-option>(B) \((x - 6)(x + 3)\)</mcq-option>
<mcq-correct>(C) \((x + 6)(x + 3)\)</mcq-correct>
<mcq-option>(D) \((x + 6)(x + 3)^2\)</mcq-option>

Question 4

<mcq-option>(A) \((6x + 3)(x + 6)\)</mcq-option>
<mcq-option>(B) \((x - 6)(z + 3)\)</mcq-option>
<mcq-correct>(C) \((x + 6)(x + 3)\)</mcq-correct>
<mcq-option>(D) \((x + 6)(x + 3)^2\)</mcq-option>
</answer>

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