Multiplying Monomials and Polynomials: Step-by-Step College Algebra Guidance

Q1. Multiply: k(k - 9)

Background

Topic: Multiplying a Monomial by a Polynomial

This question tests your ability to use the distributive property to multiply a monomial (a single term) by a polynomial (an expression with multiple terms).

Key Terms and Formulas

• Monomial: An algebraic expression with only one term (e.g., k).

• Polynomial: An expression with two or more terms (e.g., k - 9).

• Distributive Property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Identify the monomial (k) and the polynomial (k - 9).

2. Apply the distributive property: Multiply the monomial by each term in the polynomial.

3. Multiply $k \times k$ and $k \times (-9)$ separately.

4. Write the resulting terms together to form a new polynomial.

Try solving on your own before revealing the answer!

Q2. Multiply: 9a(a + 1)

Background

Topic: Distributive Property with Variables

This question checks your understanding of distributing a monomial across a binomial.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Recognize the monomial (9a) and the binomial (a + 1).

2. Multiply 9a by each term inside the parentheses: $9a \times a$ and $9a \times 1$.

3. Write out both products as separate terms.

4. Combine the terms to form the expanded expression.

Try solving on your own before revealing the answer!

Q3. Multiply: -6p(p^4 - 8)

Background

Topic: Multiplying Monomials and Polynomials with Exponents

This question involves multiplying a monomial by a binomial, where one term has an exponent.

Key Terms and Formulas

• Product Rule for Exponents: $a^m \times a^n = a^{m+n}$

• Distributive Property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Identify the monomial (-6p) and the polynomial $(p^4 - 8)$.

2. Distribute -6p to each term: $-6p \times p^4$ and $-6p \times (-8)$.

3. For $-6p \times p^4$, use the product rule for exponents.

4. Multiply $-6p$ by $-8$ for the second term.

Try solving on your own before revealing the answer!

Q4. Multiply: 3c^5(-2c - 5)

Background

Topic: Distributive Property with Negative Coefficients and Exponents

This question tests your ability to distribute a monomial with an exponent to a binomial with negative coefficients.

Key Terms and Formulas

• Product Rule for Exponents: $a^m \times a^n = a^{m+n}$

• Distributive Property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Identify the monomial $3c^5$ and the binomial $(-2c - 5)$.

2. Multiply $3c^5$ by $-2c$ using the product rule for exponents.

3. Multiply $3c^5$ by $-5$.

4. Write both terms together to form the expanded expression.

Try solving on your own before revealing the answer!

Q5. Multiply: -4g^2(2g + 7)

Background

Topic: Distributive Property with Negative Monomials

This question involves distributing a negative monomial to a binomial.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Identify the monomial $-4g^2$ and the binomial $(2g + 7)$.

2. Multiply $-4g^2$ by $2g$.

3. Multiply $-4g^2$ by $7$.

4. Combine the results to form the expanded polynomial.

Try solving on your own before revealing the answer!

Q6. Multiply: x(x^2 - x + 3)

Background

Topic: Distributive Property with Trinomials

This question tests your ability to distribute a monomial to a trinomial and use exponent rules.

Key Terms and Formulas

• Product Rule for Exponents: $a^m \times a^n = a^{m+n}$

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

Step-by-Step Guidance

1. Identify the monomial $x$ and the trinomial $(x^2 - x + 3)$.

2. Multiply $x$ by each term: $x \times x^2$, $x \times (-x)$, and $x \times 3$.

3. Apply the product rule for exponents where needed.

4. Write out the resulting terms as a single polynomial.

Try solving on your own before revealing the answer!