Dividing Polynomials by a Monomial – Step-by-Step College Algebra Guidance

Q1. Divide the polynomial $6x + 9$ by $3$.

Background

Topic: Dividing Polynomials by a Monomial

This question tests your ability to divide each term of a polynomial by a single monomial (in this case, the number 3).

Key Terms and Formulas

• Monomial: A single term, such as $3$,$x$, or $5x^2$.

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Write the expression as $\frac{6x + 9}{3}$.

2. Split the numerator so each term is divided by the monomial: $\frac{6x}{3} + \frac{9}{3}$.

3. Simplify each term separately. For $\frac{6x}{3}$, divide the coefficients and keep the variable. For $\frac{9}{3}$, divide the numbers.

Try solving on your own before revealing the answer!

Q2. Divide $10v^2 + 15v$ by $5v$.

Background

Topic: Dividing Polynomials by a Monomial

This question asks you to divide a two-term polynomial by a monomial that includes a variable.

Key Terms and Formulas

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Rewrite the expression as $\frac{10v^2 + 15v}{5v}$.

2. Separate each term: $\frac{10v^2}{5v} + \frac{15v}{5v}$.

3. For each term, divide the coefficients and apply the exponent rule for the variables.

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Q3. Divide $8a^3 + 2a^2 - 5a$ by $2a$.

Background

Topic: Dividing Polynomials by a Monomial

This problem involves dividing a three-term polynomial by a monomial with a variable.

Key Terms and Formulas

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Express the division as $\frac{8a^3 + 2a^2 - 5a}{2a}$.

2. Break it into separate terms: $\frac{8a^3}{2a} + \frac{2a^2}{2a} - \frac{5a}{2a}$.

3. For each term, divide the coefficients and subtract exponents for the variable $a$.

Try solving on your own before revealing the answer!

Q4. Divide $40x^2 - 8x$ by $8x$.

Background

Topic: Dividing Polynomials by a Monomial

This question tests your ability to divide each term of a polynomial by a monomial with a variable.

Key Terms and Formulas

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Write the division as $\frac{40x^2 - 8x}{8x}$.

2. Separate into $\frac{40x^2}{8x} - \frac{8x}{8x}$.

3. Divide coefficients and apply the exponent rule for $x$ in each term.

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Q5. Divide $18c^3 + 21c^2$ by $3c^2$.

Background

Topic: Dividing Polynomials by a Monomial

This problem involves dividing a polynomial by a monomial with an exponent.

Key Terms and Formulas

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Rewrite as $\frac{18c^3 + 21c^2}{3c^2}$.

2. Split into $\frac{18c^3}{3c^2} + \frac{21c^2}{3c^2}$.

3. Divide coefficients and subtract exponents for $c$ in each term.

Try solving on your own before revealing the answer!

Q6. Divide $6a^2b^2 - 12ab^2$ by $2ab$.

Background

Topic: Dividing Polynomials by a Monomial

This question involves dividing terms with multiple variables and exponents.

Key Terms and Formulas

• Quotient Rule for Exponents: $\frac{a^m}{a^n} = a^{m-n}$

• Apply the rule to each variable separately.

Step-by-Step Guidance

1. Write as $\frac{6a^2b^2 - 12ab^2}{2ab}$.

2. Separate: $\frac{6a^2b^2}{2ab} - \frac{12ab^2}{2ab}$.

3. Divide coefficients and subtract exponents for both $a$ and $b$ in each term.

Try solving on your own before revealing the answer!