Factoring perfect trinomial worksheet - Free Printable

Problem Analysis:

The worksheet focuses on factoring perfect square trinomials. A perfect square trinomial is of the form:
- \( (x + y)^2 = x^2 + 2xy + y^2 \)
- \( (x - y)^2 = x^2 - 2xy + y^2 \)

#### Part A: Completing the Expression to Make a Perfect Square Trinomial
For each given expression, we need to determine the missing term that makes it a perfect square trinomial. The general steps are:
1. Identify the coefficient of the linear term (\(2xy\) or \(-2xy\)).
2. Divide this coefficient by 2 to find \(y\).
3. Square \(y\) to find the constant term.

#### Part B: Matching Column A with Its Factor in Column B
Here, we need to match each perfect square trinomial in Column A with its factored form in Column B. This involves recognizing the structure of the trinomial and identifying its factorization.

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Solution:

#### Part A: Completing the Expression
We will solve each problem step by step.

1. \(a^2 + 8a + \_\_\_\)
- Coefficient of \(a\) is 8.
- Divide by 2: \(8 / 2 = 4\).
- Square the result: \(4^2 = 16\).
- Missing term: \(16\).
- Answer: \(a^2 + 8a + 16\).

2. \(a^2 - 10a + \_\_\_\)
- Coefficient of \(a\) is \(-10\).
- Divide by 2: \(-10 / 2 = -5\).
- Square the result: \((-5)^2 = 25\).
- Missing term: \(25\).
- Answer: \(a^2 - 10a + 25\).

3. \(a^2 + 4a + \_\_\_\)
- Coefficient of \(a\) is 4.
- Divide by 2: \(4 / 2 = 2\).
- Square the result: \(2^2 = 4\).
- Missing term: \(4\).
- Answer: \(a^2 + 4a + 4\).

4. \(a^2 - 24a + \_\_\_\)
- Coefficient of \(a\) is \(-24\).
- Divide by 2: \(-24 / 2 = -12\).
- Square the result: \((-12)^2 = 144\).
- Missing term: \(144\).
- Answer: \(a^2 - 24a + 144\).

5. \(a^2 - 4a + \_\_\_\)
- Coefficient of \(a\) is \(-4\).
- Divide by 2: \(-4 / 2 = -2\).
- Square the result: \((-2)^2 = 4\).
- Missing term: \(4\).
- Answer: \(a^2 - 4a + 4\).

6. \(a^2 + 24a + \_\_\_\)
- Coefficient of \(a\) is 24.
- Divide by 2: \(24 / 2 = 12\).
- Square the result: \(12^2 = 144\).
- Missing term: \(144\).
- Answer: \(a^2 + 24a + 144\).

7. \(a^2 + 12a + \_\_\_\)
- Coefficient of \(a\) is 12.
- Divide by 2: \(12 / 2 = 6\).
- Square the result: \(6^2 = 36\).
- Missing term: \(36\).
- Answer: \(a^2 + 12a + 36\).

8. \(a^2 - 12a + \_\_\_\)
- Coefficient of \(a\) is \(-12\).
- Divide by 2: \(-12 / 2 = -6\).
- Square the result: \((-6)^2 = 36\).
- Missing term: \(36\).
- Answer: \(a^2 - 12a + 36\).

9. \(a^2 - 6a + \_\_\_\)
- Coefficient of \(a\) is \(-6\).
- Divide by 2: \(-6 / 2 = -3\).
- Square the result: \((-3)^2 = 9\).
- Missing term: \(9\).
- Answer: \(a^2 - 6a + 9\).

10. \(a^2 + 10a + \_\_\_\)
- Coefficient of \(a\) is 10.
- Divide by 2: \(10 / 2 = 5\).
- Square the result: \(5^2 = 25\).
- Missing term: \(25\).
- Answer: \(a^2 + 10a + 25\).

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#### Part B: Matching Column A with Its Factor in Column B
We will match each trinomial in Column A with its corresponding factor in Column B.

1. \(a^2 + 8a + 16\)
- This is \((a + 4)^2\).
- Match: \((a + 4)^2\).

2. \(a^2 - 10a + 25\)
- This is \((a - 5)^2\).
- Match: \((a - 5)^2\).

3. \(4a^2 - 4a + 1\)
- This is \((2a - 1)^2\).
- Match: \((2a - 1)^2\).

4. \(4a^2 - 12a + 9\)
- This is \((2a - 3)^2\).
- Match: \((2a - 3)^2\).

5. \(4a^2 + 4a + 1\)
- This is \((2a + 1)^2\).
- Match: \((2a + 1)^2\).

6. \(9a^2 + 24a + 16\)
- This is \((3a + 4)^2\).
- Match: \((3a + 4)^2\).

7. \(a^2 + 12a + 36\)
- This is \((a + 6)^2\).
- Match: \((a + 6)^2\).

8. \(a^2 - 12a + 36\)
- This is \((a - 6)^2\).
- Match: \((a - 6)^2\).

9. \(a^2 - 8a + 16\)
- This is \((a - 4)^2\).
- Match: \((a - 4)^2\).

10. \(a^2 + 10a + 25\)
- This is \((a + 5)^2\).
- Match: \((a + 5)^2\).

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Final Answers:

#### Part A:
1. \(16\)
2. \(25\)
3. \(4\)
4. \(144\)
5. \(4\)
6. \(144\)
7. \(36\)
8. \(36\)
9. \(9\)
10. \(25\)

#### Part B:
1. \((a + 4)^2\)
2. \((a - 5)^2\)
3. \((2a - 1)^2\)
4. \((2a - 3)^2\)
5. \((2a + 1)^2\)
6. \((3a + 4)^2\)
7. \((a + 6)^2\)
8. \((a - 6)^2\)
9. \((a - 4)^2\)
10. \((a + 5)^2\)

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Boxed Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{Part A:} & 16, 25, 4, 144, 4, 144, 36, 36, 9, 25 \\
\text{Part B:} & (a + 4)^2, (a - 5)^2, (2a - 1)^2, (2a - 3)^2, (2a + 1)^2, (3a + 4)^2, (a + 6)^2, (a - 6)^2, (a - 4)^2, (a + 5)^2
\end{array}
}
\]