Factoring Common Factors In Algebra

1) Factor the common factor out of the expression: 12b² – 18

6(2b² – 3)

3(4b² – 6)

2(6b² – 9)

6(2b² + 3)

To factor the expression 12b² – 18, we first identify the greatest common factor (GCF) of the coefficients 12 and 18, which is 6. We then divide each term in the expression by this GCF. Dividing 12b² by 6 gives us 2b², and dividing -18 by 6 gives us -3. Thus, we can rewrite the expression as 6(2b² - 3), effectively factoring out the common factor. This simplifies the expression while maintaining its equivalence.

Explanation

To factor the expression 12b² – 18, we first identify the greatest common factor (GCF) of the coefficients 12 and 18, which is 6. We then divide each term in the expression by this GCF. Dividing 12b² by 6 gives us 2b², and dividing -18 by 6 gives us -3. Thus, we can rewrite the expression as 6(2b² - 3), effectively factoring out the common factor. This simplifies the expression while maintaining its equivalence.

2) Factor the common factor out of the expression: –12a + 6

3(–4a + 2)

6(–2a + 1)

2(–6a + 3)

6(2a – 1)

To factor the expression –12a + 6, first identify the greatest common factor (GCF) of the terms. The GCF of –12a and 6 is 6. By dividing both terms by 6, we can rewrite the expression as 6 multiplied by the simplified form of the remaining terms. This gives us 6(–2a + 1), where –2a and 1 are the results of dividing –12a and 6 respectively by 6. Thus, the expression is factored correctly.

Explanation

To factor the expression –12a + 6, first identify the greatest common factor (GCF) of the terms. The GCF of –12a and 6 is 6. By dividing both terms by 6, we can rewrite the expression as 6 multiplied by the simplified form of the remaining terms. This gives us 6(–2a + 1), where –2a and 1 are the results of dividing –12a and 6 respectively by 6. Thus, the expression is factored correctly.

3) Factor the common factor out of: –14a⁷ + 49a⁵ – 63a³

7a²(–2a⁵ + 7a³ – 9a)

7a³(–2a⁴ + 7a² – 9)

14a³(–a⁴ + 3a² – 4)

7a(–2a⁶ + 7a⁴ – 9a²)

To factor the expression –14a⁷ + 49a⁵ – 63a³, we first identify the greatest common factor (GCF) from the coefficients and the variable terms. The coefficients –14, 49, and –63 have a GCF of 7, and the variable part has a minimum power of a³. Thus, the GCF is 7a³. Factoring this out, we rewrite the expression as 7a³(–2a⁴ + 7a² – 9), simplifying the polynomial inside the parentheses. This process highlights how to systematically extract common factors to simplify algebraic expressions.

Explanation

To factor the expression –14a⁷ + 49a⁵ – 63a³, we first identify the greatest common factor (GCF) from the coefficients and the variable terms. The coefficients –14, 49, and –63 have a GCF of 7, and the variable part has a minimum power of a³. Thus, the GCF is 7a³. Factoring this out, we rewrite the expression as 7a³(–2a⁴ + 7a² – 9), simplifying the polynomial inside the parentheses. This process highlights how to systematically extract common factors to simplify algebraic expressions.

4) Factor the common factor out of: 45v² + 63v³ – 9v⁴

3v²(15 + 21v – 3v²)

9v(5v + 7v² – v³)

9v²(5 + 7v – v²)

9(5v² + 7v³ – v⁴)

To factor the expression 45v² + 63v³ – 9v⁴, we first identify the greatest common factor (GCF) of the coefficients (45, 63, and -9), which is 9, and the lowest power of v, which is v². Factoring out 9v² from each term results in the expression 9v²(5 + 7v - v²). This simplifies the original polynomial while highlighting the common factor, making it easier to work with or further analyze.

Explanation

To factor the expression 45v² + 63v³ – 9v⁴, we first identify the greatest common factor (GCF) of the coefficients (45, 63, and -9), which is 9, and the lowest power of v, which is v². Factoring out 9v² from each term results in the expression 9v²(5 + 7v - v²). This simplifies the original polynomial while highlighting the common factor, making it easier to work with or further analyze.

5) Factor the common factor out of: 63a⁶ – 9a³ – 54a + 36

9(7a⁶ – a³ – 6a + 4)

3(21a⁶ – 3a³ – 18a + 12)

9a(7a⁵ – a² – 6 + 4/a)

9(7a⁵ – a² – 6a + 4)

To factor out the common factor from the expression 63a⁶ – 9a³ – 54a + 36, we first identify the greatest common factor (GCF) of the coefficients, which is 9. Dividing each term by 9 simplifies the expression to 7a⁶ – a³ – 6a + 4. Thus, factoring out 9 gives us 9(7a⁶ – a³ – 6a + 4), which is the factored form of the original expression. This shows how the GCF can simplify polynomial expressions effectively.

Explanation

To factor out the common factor from the expression 63a⁶ – 9a³ – 54a + 36, we first identify the greatest common factor (GCF) of the coefficients, which is 9. Dividing each term by 9 simplifies the expression to 7a⁶ – a³ – 6a + 4. Thus, factoring out 9 gives us 9(7a⁶ – a³ – 6a + 4), which is the factored form of the original expression. This shows how the GCF can simplify polynomial expressions effectively.

6) Factor the common factor out of: 3a⁶ + 6a⁴ – 30a² + 6a

3a(a⁵ + 2a³ – 10a + 2)

6a(a⁵ + a³ – 5a + 1)

3a²(a⁴ + 2a² – 10 + 2/a)

3(a⁶ + 2a⁴ – 10a² + 2a)

To factor out the common factor from the expression 3a⁶ + 6a⁴ – 30a² + 6a, we first identify the greatest common factor (GCF) of all terms. The GCF here is 3a, as it is the highest coefficient and the lowest power of 'a' present in each term. Dividing each term by 3a gives us the remaining polynomial: a⁵ + 2a³ – 10a + 2. Thus, the expression can be rewritten as 3a(a⁵ + 2a³ – 10a + 2).

Explanation

To factor out the common factor from the expression 3a⁶ + 6a⁴ – 30a² + 6a, we first identify the greatest common factor (GCF) of all terms. The GCF here is 3a, as it is the highest coefficient and the lowest power of 'a' present in each term. Dividing each term by 3a gives us the remaining polynomial: a⁵ + 2a³ – 10a + 2. Thus, the expression can be rewritten as 3a(a⁵ + 2a³ – 10a + 2).

7) Factor the common factor out of: 9xy² + 9xy + 15x²y²

3xy(3y + 3 + 5xy)

9xy(y + 1 + 5xy/3)

3x(3y² + 3y + 5xy²)

3y(3xy + 3x + 5x²y)

To factor out the common factor from the expression 9xy² + 9xy + 15x²y², we first identify the greatest common factor (GCF) of all the terms. The GCF is 3xy, which can be factored out, leading to the expression 3xy(3y + 3 + 5xy). This shows that each term in the original expression can be expressed as a multiple of 3xy, simplifying the equation and making it easier to work with.

Explanation

To factor out the common factor from the expression 9xy² + 9xy + 15x²y², we first identify the greatest common factor (GCF) of all the terms. The GCF is 3xy, which can be factored out, leading to the expression 3xy(3y + 3 + 5xy). This shows that each term in the original expression can be expressed as a multiple of 3xy, simplifying the equation and making it easier to work with.

8) Factor the common factor out of: 20m⁵n³ + 8m²n² + 36m²n

4m²n(5m³n² + 2n + 9)

4mn(5m⁴n² + 2mn + 9m)

2m²n(10m³n² + 4n + 18)

4m²n²(5m³n + 2 + 9/n)

To factor the given expression, first identify the greatest common factor (GCF) of the terms: 20m⁵n³, 8m²n², and 36m²n. The GCF is 4m²n. Dividing each term by this factor yields the simplified expression: 5m³n² + 2n + 9. Therefore, the factored form is 4m²n(5m³n² + 2n + 9), which clearly shows the common factor extracted from the original polynomial.

Explanation

To factor the given expression, first identify the greatest common factor (GCF) of the terms: 20m⁵n³, 8m²n², and 36m²n. The GCF is 4m²n. Dividing each term by this factor yields the simplified expression: 5m³n² + 2n + 9. Therefore, the factored form is 4m²n(5m³n² + 2n + 9), which clearly shows the common factor extracted from the original polynomial.