Factoring Polynomials Using the Greatest Common Factor (GCF)

Q1. Simplify the polynomial: 3a + 7a

Background

Topic: Combining Like Terms in Polynomials

This question tests your ability to identify and combine like terms in a polynomial expression, which is a foundational skill in algebra.

Key Terms and Formulas:

• Like Terms: Terms that have the same variable(s) raised to the same power(s).

• Combining Like Terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Identify the like terms in the expression. Both terms have the variable $a$.

2. Add the coefficients of $a$: $3a + 7a$.

3. Write the result as a single term with the combined coefficient.

Try solving on your own before revealing the answer!

Q2. Simplify the polynomial: 2m^2 + 6m - m^2 - 1

Background

Topic: Combining Like Terms in Polynomials

This question asks you to simplify a polynomial by combining like terms, which involves grouping terms with the same variable and exponent.

Key Terms and Formulas:

• Like Terms: Terms with the same variable(s) and exponent(s).

• Combining Like Terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Group the $m^2$ terms together: $2m^2$ and $-m^2$.

2. Group the $m$ term: $+6m$.

3. Include the constant term: $-1$.

4. Combine the $m^2$ terms: $2m^2 - m^2$.

5. Write the simplified expression by combining all terms.

Try solving on your own before revealing the answer!

Q3. Simplify the polynomial: 3x^2y - 4x^2y

Background

Topic: Combining Like Terms in Polynomials

This question tests your ability to combine like terms with multiple variables.

Key Terms and Formulas:

• Like Terms: Both terms have $x^2y$.

• Combining Like Terms: Add or subtract the coefficients.

Step-by-Step Guidance

1. Identify the like terms: $3x^2y$ and $-4x^2y$.

2. Subtract the coefficients: $3 - 4$.

3. Write the result as a single term with the combined coefficient.

Try solving on your own before revealing the answer!

Q4. Factor the polynomial: 3x + 12

Background

Topic: Factoring Polynomials Using the Greatest Common Factor (GCF)

This question asks you to factor out the GCF from a two-term polynomial.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest factor that divides each term of the polynomial.

• Factoring: Expressing the polynomial as a product of its GCF and another polynomial.

Step-by-Step Guidance

1. Identify the GCF of the coefficients: $3$and $12$.

2. Check if the variable $x$ is present in both terms (it's only in one).

3. Factor out the GCF from both terms.

4. Write the expression as the product of the GCF and the remaining terms in parentheses.

Try solving on your own before revealing the answer!

Q5. Factor the polynomial: 7y - 7

Background

Topic: Factoring Polynomials Using the Greatest Common Factor (GCF)

This question asks you to factor out the GCF from a two-term polynomial.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest factor that divides each term of the polynomial.

• Factoring: Expressing the polynomial as a product of its GCF and another polynomial.

Step-by-Step Guidance

1. Identify the GCF of the coefficients: $7$and$-7$.

2. Check if the variable $y$ is present in both terms (it's only in one).

3. Factor out the GCF from both terms.

4. Write the expression as the product of the GCF and the remaining terms in parentheses.

Try solving on your own before revealing the answer!

Q6. Factor the polynomial: 8m + 36n

Background

Topic: Factoring Polynomials Using the Greatest Common Factor (GCF)

This question asks you to factor out the GCF from a two-term polynomial with different variables.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest factor that divides each term of the polynomial.

• Factoring: Expressing the polynomial as a product of its GCF and another polynomial.

Step-by-Step Guidance

1. Identify the GCF of the coefficients: $8$and $36$.

2. Check if any variables are present in both terms (in this case, $m$ and $n$ are different).

3. Factor out the GCF from both terms.

4. Write the expression as the product of the GCF and the remaining terms in parentheses.

Try solving on your own before revealing the answer!

Q7. Factor the polynomial: 5x + 30y

Background

Topic: Factoring Polynomials Using the Greatest Common Factor (GCF)

This question asks you to factor out the GCF from a two-term polynomial with different variables.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest factor that divides each term of the polynomial.

• Factoring: Expressing the polynomial as a product of its GCF and another polynomial.

Step-by-Step Guidance

1. Identify the GCF of the coefficients: $5$and $30$.

2. Check if any variables are present in both terms (in this case, $x$ and $y$ are different).

3. Factor out the GCF from both terms.

4. Write the expression as the product of the GCF and the remaining terms in parentheses.

Try solving on your own before revealing the answer!