Question

factor \\(x^2 - 3x - 28\\).

\\(ax^2 + bx + c\\)

identify the values that should be written to complete the x diagram.
on the top:
on the bottom:
on the side:

rewrite the expression using the numbers on the sides of the diagram.

use double grouping to factor the four terms.
\\(x^2 - 3x - 28 =\\)

Explanation:

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<reasoning>

Identify coefficients of the trinomial

For the trinomial \(x^2 - 3x - 28\), we compare it to the standard form \(ax^2 + bx + c\):
\[
a = 1,\quad b = -3,\quad c = -28
\]

Determine values for the X diagram

The X diagram template shows:

• Top value: \(ac = 1 \cdot (-28) = -28\)
• Bottom value: \(b = -3\)
• Side values: Two numbers that multiply to \(ac = -28\) and add to \(b = -3\). These numbers are \(-7\) and \(4\).

Rewrite and factor by grouping

Rewrite the middle term \(-3x\) using the side numbers \(-7\) and \(4\):
\[
x^2 - 7x + 4x - 28
\]
Group the terms:
\[
(x^2 - 7x) + (4x - 28) = x(x - 7) + 4(x - 7) = (x - 7)(x + 4)
\]
</reasoning>

<answer>

Question 1

Identify the values that should be written to complete the X diagram.

• On the top: <blank>-28</blank>
• On the bottom: <blank>-3</blank>
• On the sides: <blank>-7</blank> and <blank>4</blank>

Question 2

Rewrite the expression using the numbers on the sides of the X diagram.
<blank>\(x^2 - 7x + 4x - 28\)</blank>

Question 3

Use double grouping to factor the four terms.
\(x^2 - 3x - 28 =\) <blank>\((x - 7)(x + 4)\)</blank>
</answer>

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Answer:

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<reasoning>

Identify coefficients of the trinomial

For the trinomial \(x^2 - 3x - 28\), we compare it to the standard form \(ax^2 + bx + c\):
\[
a = 1,\quad b = -3,\quad c = -28
\]

Determine values for the X diagram

The X diagram template shows:

• Top value: \(ac = 1 \cdot (-28) = -28\)
• Bottom value: \(b = -3\)
• Side values: Two numbers that multiply to \(ac = -28\) and add to \(b = -3\). These numbers are \(-7\) and \(4\).

Rewrite and factor by grouping

Rewrite the middle term \(-3x\) using the side numbers \(-7\) and \(4\):
\[
x^2 - 7x + 4x - 28
\]
Group the terms:
\[
(x^2 - 7x) + (4x - 28) = x(x - 7) + 4(x - 7) = (x - 7)(x + 4)
\]
</reasoning>

<answer>

Question 1

Identify the values that should be written to complete the X diagram.

• On the top: <blank>-28</blank>
• On the bottom: <blank>-3</blank>
• On the sides: <blank>-7</blank> and <blank>4</blank>

Question 2

Rewrite the expression using the numbers on the sides of the X diagram.
<blank>\(x^2 - 7x + 4x - 28\)</blank>

Question 3

Use double grouping to factor the four terms.
\(x^2 - 3x - 28 =\) <blank>\((x - 7)(x + 4)\)</blank>
</answer>

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"Factoring Polynomials"
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