Question

consider the trinomial \\(9x^2 + 21x + 10\\).

what value is placed on top of the x?
what value is placed on the bottom of the x?
what is the factored form of \\(9x^2 + 21x + 10\\)?

Explanation:

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<reasoning>

Identify coefficients of the trinomial

\[

$$\begin{aligned} &9x^2 + 21x + 10 \implies a = 9,\ b = 21,\ c = 10 \end{aligned}$$

\]

Calculate values for the X method diagram

\[

$$\begin{aligned} &\text{Top value: } ac = 9 \times 10 = 90\\ &\text{Bottom value: } b = 21 \end{aligned}$$

\]

Find factors and write the factored form

\[

$$\begin{aligned} &\text{Find two numbers that multiply to } 90 \text{ and add to } 21: 15 \text{ and } 6\\ &9x^2 + 21x + 10 = 9x^2 + 15x + 6x + 10 = 3x(3x + 5) + 2(3x + 5) = (3x + 5)(3x + 2) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

What value is placed on top of the X? <blank>90</blank>

Question 2

What value is placed on the bottom of the X? <blank>21</blank>

Question 3

What is the factored form of \( 9x^2 + 21x + 10 \)? <blank>\((3x + 5)(3x + 2)\)</blank>
</answer>

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"Mathematics",
"Algebra",
"Factoring Polynomials"
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Answer:

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<reasoning>

Identify coefficients of the trinomial

\[

$$\begin{aligned} &9x^2 + 21x + 10 \implies a = 9,\ b = 21,\ c = 10 \end{aligned}$$

\]

Calculate values for the X method diagram

\[

$$\begin{aligned} &\text{Top value: } ac = 9 \times 10 = 90\\ &\text{Bottom value: } b = 21 \end{aligned}$$

\]

Find factors and write the factored form

\[

$$\begin{aligned} &\text{Find two numbers that multiply to } 90 \text{ and add to } 21: 15 \text{ and } 6\\ &9x^2 + 21x + 10 = 9x^2 + 15x + 6x + 10 = 3x(3x + 5) + 2(3x + 5) = (3x + 5)(3x + 2) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

What value is placed on top of the X? <blank>90</blank>

Question 2

What value is placed on the bottom of the X? <blank>21</blank>

Question 3

What is the factored form of \( 9x^2 + 21x + 10 \)? <blank>\((3x + 5)(3x + 2)\)</blank>
</answer>

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"Factoring Polynomials"
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