Step-by-Step Guidance: Multiplying Binomials and Trinomials in College Algebra

Q1. Multiply the binomials: (y + 8)(y + 1)

Background

Topic: Multiplying Binomials (FOIL Method)

This question tests your ability to multiply two binomials using the distributive property, often called the FOIL method (First, Outer, Inner, Last).

Key Terms and Formulas

• Binomial: A polynomial with two terms (e.g., y + 8).

• FOIL Method: Multiply the First, Outer, Inner, and Last terms, then combine like terms.

General formula: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms in each binomial: $(y + 8)$ and $(y + 1)$.

2. Apply the distributive property (FOIL):

   • First: Multiply the first terms: $y \times y$

   • Outer: Multiply the outer terms: $y \times 1$

   • Inner: Multiply the inner terms: $8 \times y$

   • Last: Multiply the last terms: $8 \times 1$

3. Write out each product as a separate term.

4. Combine like terms where possible.

Try solving on your own before revealing the answer!

Q2. Multiply the binomials: (k – 4)(k + 5)

Background

Topic: Multiplying Binomials (FOIL Method)

This question also uses the distributive property to multiply two binomials.

Key Terms and Formulas

• FOIL Method: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms: $(k - 4)$ and $(k + 5)$.

2. Multiply each term in the first binomial by each term in the second binomial:

   • First: $k \times k$

   • Outer: $k \times 5$

   • Inner: $-4 \times k$

   • Last: $-4 \times 5$

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q3. Multiply the binomials: (x – 10)(x – 4)

Background

Topic: Multiplying Binomials (FOIL Method)

This problem reinforces the distributive property and combining like terms.

Key Terms and Formulas

• FOIL Method: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms: $(x - 10)$ and $(x - 4)$.

2. Multiply each term in the first binomial by each term in the second binomial:

   • First: $x \times x$

   • Outer: $x \times -4$

   • Inner: $-10 \times x$

   • Last: $-10 \times -4$

3. Write out all four terms.

4. Combine like terms to simplify.

Try solving on your own before revealing the answer!

Q4. Multiply the binomials: (x + 2)(x – 2)

Background

Topic: Special Products – Difference of Squares

This is a special case where the binomials are conjugates. Multiplying them results in a difference of squares.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Recognize the pattern: $(x + 2)(x - 2)$ is of the form $(a + b)(a - b)$.

2. Apply the difference of squares formula: $a^2 - b^2$.

3. Identify $a = x$ and $b = 2$.

4. Substitute into the formula and simplify.

Try solving on your own before revealing the answer!

Q5. Multiply the binomials: (4x – 7)(x + 3)

Background

Topic: Multiplying Binomials (Distributive Property)

This question requires distributing each term in the first binomial to each term in the second binomial.

Key Terms and Formulas

• Distributive Property: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms: $(4x - 7)$ and $(x + 3)$.

2. Multiply $4x$ by both $x$ and $3$.

3. Multiply $-7$ by both $x$ and $3$.

4. Write out all four terms and combine like terms.

Try solving on your own before revealing the answer!

Q6. Multiply the binomials: (n – 1)(5n – 4)

Background

Topic: Multiplying Binomials

This problem reinforces the distributive property and combining like terms.

Key Terms and Formulas

• Distributive Property: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms: $(n - 1)$ and $(5n - 4)$.

2. Multiply $n$ by $5n$ and $-4$.

3. Multiply $-1$ by $5n$ and $-4$.

4. Write out all four terms and combine like terms.

Try solving on your own before revealing the answer!

Q7. Multiply the binomials: (3y + 1)(3y + 2)

Background

Topic: Multiplying Binomials

This question tests your ability to multiply binomials with coefficients greater than 1.

Key Terms and Formulas

• Distributive Property: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms: $(3y + 1)$ and $(3y + 2)$.

2. Multiply $3y$ by $ 3y $ and $2$.

3. Multiply $1$ by $3y$ and $2$.

4. Write out all four terms and combine like terms.

Try solving on your own before revealing the answer!

Q8. Multiply the binomials: (6a + 2)(2a + 3)

Background

Topic: Multiplying Binomials

This problem involves multiplying binomials with coefficients and combining like terms.

Key Terms and Formulas

• Distributive Property: $(a + b)(c + d) = ac + ad + bc + bd$

Step-by-Step Guidance

1. Identify the terms: $(6a + 2)$ and $(2a + 3)$.

2. Multiply $6a$ by $2a$ and $3$.

3. Multiply $2$ by $2a$ and $3$.

4. Write out all four terms and combine like terms.

Try solving on your own before revealing the answer!