Factoring and Simplifying Polynomials: Difference of Squares & GCF

Q1. Simplify: (x + 4)(x - 4)

Background

Topic: Multiplying Binomials / Difference of Squares

This question tests your ability to multiply two binomials and recognize the pattern known as the difference of squares.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

• FOIL Method: First, Outer, Inner, Last (for multiplying binomials)

Step-by-Step Guidance

1. Identify the binomials: $(x + 4)$ and $(x - 4)$.

2. Apply the FOIL method to multiply: $(x + 4)(x - 4)$.

3. Multiply the first terms: $x \times x$.

4. Multiply the outer and inner terms: $x \times -4$ and $4 \times x$.

5. Multiply the last terms: $4 \times -4$.

Try solving on your own before revealing the answer!

Q2. Simplify: (m + 5)(m - 5)

Background

Topic: Multiplying Binomials / Difference of Squares

This question is similar to Q1, testing your ability to multiply binomials and recognize the difference of squares pattern.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Identify the binomials: $(m + 5)$ and $(m - 5)$.

2. Apply the FOIL method to multiply: $(m + 5)(m - 5)$.

3. Multiply the first terms: $m \times m$.

4. Multiply the outer and inner terms: $m \times -5$ and $5 \times m$.

5. Multiply the last terms: $5 \times -5$.

Try solving on your own before revealing the answer!

Q3. Simplify: (a + 2b)(a - 2b)

Background

Topic: Multiplying Binomials / Difference of Squares

This question tests your ability to multiply binomials with variables and coefficients, and recognize the difference of squares pattern.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Identify the binomials: $(a + 2b)$ and $(a - 2b)$.

2. Apply the FOIL method to multiply: $(a + 2b)(a - 2b)$.

3. Multiply the first terms: $a \times a$.

4. Multiply the outer and inner terms: $a \times -2b$ and $2b \times a$.

5. Multiply the last terms: $2b \times -2b$.

Try solving on your own before revealing the answer!

Q4. What is the factoring rule for a difference of squares?

Background

Topic: Factoring Polynomials

This question asks you to recall the general rule for factoring expressions that are a difference of squares.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Recognize that the expression must be a subtraction of two perfect squares.

2. Recall the formula: $a^2 - b^2 = (a + b)(a - b)$.

3. Apply this rule to any expression that fits the pattern.

Try stating the rule in your own words before checking the answer!

Q5. Factor: a^2 - 4

Background

Topic: Factoring Difference of Squares

This question tests your ability to factor a binomial that is a difference of squares.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Identify if both terms are perfect squares: $a^2$ and $4$.

2. Write $4$as $2^2$ to see the pattern clearly.

3. Apply the difference of squares formula: $a^2 - 2^2 = (a + 2)(a - 2)$.

Try factoring before revealing the answer!

Q6. Factor: n^2 - 64

Background

Topic: Factoring Difference of Squares

This question tests your ability to factor a binomial that is a difference of squares.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Identify if both terms are perfect squares: $n^2$ and $64$.

2. Write $64$as $8^2$ to see the pattern clearly.

3. Apply the difference of squares formula: $n^2 - 8^2 = (n + 8)(n - 8)$.

Try factoring before revealing the answer!

Q7. Factor: 81 - x^2

Background

Topic: Factoring Difference of Squares

This question tests your ability to factor a binomial that is a difference of squares.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Identify if both terms are perfect squares: $81$and$x^2$.

2. Write $81$as $9^2$ to see the pattern clearly.

3. Apply the difference of squares formula: $9^2 - x^2 = (9 + x)(9 - x)$.

Try factoring before revealing the answer!

Q8. Factor: c^2 - 100

Background

Topic: Factoring Difference of Squares

This question tests your ability to factor a binomial that is a difference of squares.

Key Terms and Formulas

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

Step-by-Step Guidance

1. Identify if both terms are perfect squares: $c^2$ and $100$.

2. Write $100$as $10^2$ to see the pattern clearly.

3. Apply the difference of squares formula: $c^2 - 10^2 = (c + 10)(c - 10)$.

Try factoring before revealing the answer!

Q9. Factor: k^2 + 25

Background

Topic: Factoring Polynomials

This question tests your ability to recognize when a binomial is not a difference of squares (since it is a sum, not a difference).

Key Terms and Formulas

• Difference of Squares applies only to subtraction: $a^2 - b^2$.

Step-by-Step Guidance

1. Check if both terms are perfect squares: $k^2$ and $25$.

2. Notice that the expression is a sum, not a difference.

3. Recall that the sum of squares cannot be factored using real numbers.

Try to determine if this is prime before revealing the answer!