Factor the GCF out of: 8 x + 12 8x+12

4 ( 2 x + 3 ) 4\left(2x+3\right)

2 ( 4 x + 6 ) 2\left(4x+6\right)

2 ( 4 x + 3 ) 2\left(4x+3\right)

4 ( 2 x + 6 ) 4\left(2x+6\right)

Factor the GCF out of: 6 x 2 + 9 x 6x^2+9x

3 x ( 2 x + 3 ) 3x\left(2x+3\right)

3 ( 2 x 2 + 3 x ) 3\left(2x^2+3x\right)

6 x ( x + 3 ) 6x\left(x+3\right)

3 x ( 2 x + 6 ) 3x\left(2x+6\right)

Factor the GCF from the polynomial. 18 x 3 y 2 − 27 x 2 y 3 + 9 x 4 y 18x^3y^2-27x^2y^3+9x^4y

9 x 2 y ( 2 x y − 3 y 2 + x 2 ) 9x^2y\left(2xy-3y^2+x^2\right)

9 x 2 y ( 2 x 2 − 3 x y + 3 x 2 ) 9x^2y\left(2x^2-3xy+3x^2\right)

9 x 2 y ( 2 x y + 3 y 2 + x 2 ) 9x^2y\left(2xy+3y^2+x^2\right)

9 x 2 y ( 2 x 2 − 3 x y + x 3 ) 9x^2y\left(2x^2-3xy+x^3\right)

Use grouping to factor out the polynomial. x y + 2 x + 3 y + 6 xy+2x+3y+6

( x + 3 ) ( y + 2 ) \left(x+3\right)\left(y+2\right)

( x + 2 ) ( y + 3 ) \left(x+2\right)\left(y+3\right)

( x + 3 ) ( y + 6 ) \left(x+3\right)\left(y+6\right)

( x + 2 ) ( y + 6 ) \left(x+2\right)\left(y+6\right)

7. Factoring

Factoring by Greatest Common Factor & Grouping

7. Factoring

Factoring by Greatest Common Factor & Grouping: Videos & Practice Problems

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Topic summary

Factoring polynomials involves rewriting expressions as products of factors, starting with identifying the greatest common factor (GCF) by prime factorization of coefficients and variables. For example, the GCF of $9t^2$ and $54t$ is $9t$ . Factoring by grouping applies to four-term polynomials by pairing terms and factoring out common binomials. Techniques like FOIL verify factored forms. Understanding terms, coefficients, exponents, and polynomial degrees is essential for simplifying multivariable polynomials and preparing for advanced factoring methods.

concept

GCF of Terms

Video duration:

GCF of Terms Video Summary

Factoring polynomials is a fundamental skill in algebra that involves rewriting expressions as products of simpler terms. A crucial step in this process is understanding the Greatest Common Factor (GCF), which is the largest factor shared by all terms in a given expression. The GCF is found by identifying the product of all common prime factors among the numbers or terms involved.

To find the GCF of numbers, start by expressing each number in its prime factorization. For example, the number 9 factors into $3 \times 3$, while 54 factors into $3 \times 3 \times 2 \times 3$. The common prime factors between 9 and 54 are two 3s, so the GCF is $3 \times 3 = 9$.

When applying this to algebraic terms, factor both the numerical coefficients and the variables. For instance, consider the terms \$9t^2$ and $54t$. The factorizations are \(3 \times 3 \times t \times t$ and $3 \times 3 \times 2 \times 3 \times t$, respectively. The common factors include two 3s and one \)t$, so the GCF is \(3 \times 3 \times t = 9t$.

For more complex expressions, such as \)12a^2$, $30a^3$, and $42a^5$, break down each term into prime factors and variables: \(12a^2 = 2 \times 2 \times 3 \times a \times a$, $30a^3 = 3 \times 2 \times 5 \times a \times a \times a$, and $42a^5 = 2 \times 3 \times 7 \times a \times a \times a \times a \times a$. The common factors are \)2$, $3$, and $a^2$, so the GCF is $2 \times 3 \times a^2 = 6a^2$.

Another way to understand the GCF is as the largest factor that divides each term evenly, both numerically and in terms of variable powers. Recognizing this helps simplify expressions and is essential for factoring polynomials effectively. Mastery of finding the GCF lays the groundwork for solving polynomial equations and simplifying algebraic expressions.

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Problem

What is the greatest common factor of the following lists of numbers?
$42 comma 70 comma 98$

$14$

$2$

$1470$

$42$

Problem

What is the greatest common factor of the following lists of numbers?
$19 comma 57$

$3$

$57$

$19$

$1$

example

GCF of Terms Example 1

Video duration:

46s

GCF of Terms Example 1 Video Summary

To find the greatest common factor (GCF) of the monomials, start by expressing each term as a product of its prime factors. For the term a² b, this breaks down into a × a × b. The term a b³ can be written as a × b × b × b. Lastly, a³ b² expands to a × a × a × b × b.

By comparing these factorizations, identify the common factors present in all terms. Each monomial contains at least one factor of a and one factor of b. Therefore, the greatest common factor is the product of these shared factors, which is a × b, or simply ab.

Problem

What is the greatest common factor of the following sets of monomials?
$6 x comma 9 x squared comma 15 x cubed$

$x$

$3 x$

$15 x$

$x^{3}$

Problem

What is the greatest common factor of the following sets of monomials?
$30 m^{3} n^{2}, 45 m^{2} n^{4}, 75 m^{4} n^{3}$

$5 m squared n squared$

$15 m squared n squared$

$450 m to the fourth power n cubed$

$15$

concept

Factoring the GCF out of Polynomials

Video duration:

Factoring the GCF out of Polynomials Video Summary

Factoring polynomials is a fundamental skill in algebra that involves breaking down expressions into simpler components. One of the most essential techniques is factoring by the greatest common factor (GCF). The GCF is defined as the largest factor that divides evenly into every term of a polynomial. To find the GCF, start by expressing each term as a product of its prime factors and variables, then identify the common factors shared by all terms.

For example, consider the terms 9𝑡² and 54𝑡. The prime factorization of 9𝑡² is 3 × 3 × 𝑡 × 𝑡, and for 54𝑡 it is 2 × 3 × 3 × 3 × 𝑡. The common factors are two 3s and one 𝑡, so the GCF is 3 × 3 × 𝑡 = 9𝑡. Factoring out the GCF involves rewriting each term as a product of the GCF and another factor. For 9𝑡², this is 9𝑡 × 𝑡, and for 54𝑡, it is 9𝑡 × 6. Thus, the polynomial can be expressed as:

\[9t^2 - 54t = 9t(t - 6)\]

This process is essentially reverse distribution, where you factor out the common term and simplify the remaining expression inside the parentheses. To verify the factorization, multiply the GCF back through the terms to ensure the original polynomial is recovered.

Another example involves the polynomial 6𝑥 + 12𝑥³ − 24𝑥⁴. Begin by finding the prime factorization of each term:

6𝑥 = 2 × 3 × 𝑥
12𝑥³ = 2 × 2 × 3 × 𝑥 × 𝑥 × 𝑥
24𝑥⁴ = 2 × 2 × 2 × 3 × 𝑥 × 𝑥 × 𝑥 × 𝑥

The common factors across all terms are 2, 3, and one 𝑥, so the GCF is 6𝑥. Rewriting each term as a product of 6𝑥 and another factor gives:

6𝑥 = 6𝑥 × 1
12𝑥³ = 6𝑥 × 2𝑥²
24𝑥⁴ = 6𝑥 × 4𝑥³

Factoring out 6𝑥, the polynomial becomes:

\[6x + 12x^3 - 24x^4 = 6x(1 + 2x^2 - 4x^3)\]

Factoring by the greatest common factor simplifies polynomials and often serves as the first step before applying more advanced factoring techniques. Recognizing when all terms share common factors, such as all being even or multiples of a certain term, helps quickly identify the GCF. This method not only streamlines expressions but also lays the groundwork for solving polynomial equations and simplifying algebraic fractions.

Problem

Factor the GCF out of:
$8 x plus 12$

$2 (4 x + 6)$

$2 (4 x + 3)$

$4 (2 x + 3)$

$4 (2 x + 6)$

example

Factoring the GCF out of Polynomials Example 2

Video duration:

Factoring the GCF out of Polynomials Example 2 Video Summary

Factoring polynomials with multiple variables can seem challenging, but using the Greatest Common Factor (GCF) method simplifies the process. To begin, identify the GCF by breaking down each term into its prime factors. For example, consider a polynomial with terms involving variables a and b. The first term might factor into 2 × 5 × 3 a³ × b, the second into 3 × 5 × 2 a² × 2 b², and the third into 5 × 5 × a × 3 b³.

Next, determine the common factors across all terms. Here, each term contains a factor of 5, at least one a, and at least one b. Therefore, the GCF is 5 × a × b. Rewriting each term as a product of the GCF and another factor helps in factoring the polynomial:

To get the first term from the GCF, multiply by 2 × a².
For the second term, multiply by 3 × a × b.
For the third term, multiply by 5 × b².

Factoring out the GCF from the polynomial results in:

\[5ab(2a^{2} + 3ab + 5b^{2})\]

This expression represents the fully factored form of the original polynomial. Understanding how to identify and extract the greatest common factor is essential for simplifying complex polynomials, especially those involving multiple variables. This method not only streamlines the factoring process but also lays the foundation for more advanced algebraic techniques.

Problem

Factor the GCF out of:
$6 x squared plus 9 x$

$3 x (2 x + 3)$

$3 (2 x^{2} + 3 x)$

$6 x (x + 3)$

$3 x (2 x + 6)$

Problem

Factor the GCF from the polynomial.
$18 x^{3} y^{2} - 27 x^{2} y^{3} + 9 x^{4} y$

$9 x^{2} y (2 x^{2} - 3 x y + 3 x^{2})$

$9 x^{2} y (2 x y + 3 y^{2} + x^{2})$

$9 x^{2} y (2 x^{2} - 3 x y + x^{3})$

$9 x^{2} y (2 x y - 3 y^{2} + x^{2})$

example

Factoring the GCF out of Polynomials Example 3

Video duration:

58s

Factoring the GCF out of Polynomials Example 3 Video Summary

To factor a polynomial that may initially seem different, you can apply the greatest common factor (GCF) method effectively. Begin by identifying the factors of each term in the polynomial. For example, if one term factors into 3, x, and (x + 4), and the other term factors into 5 and (x + 4), the common factor between both terms is (x + 4). Recognizing this common binomial factor allows you to factor it out of the entire polynomial. After factoring out (x + 4), the remaining expression inside the parentheses is formed by dividing each original term by the GCF, resulting in (3x - 5). Thus, the polynomial is factored as the product of the greatest common factor and the simplified binomial:

\[(x + 4)(3x - 5)\]

concept

Factoring Polynomials by Grouping

Video duration:

Factoring Polynomials by Grouping Video Summary

Factoring polynomials is a fundamental skill in algebra, especially when the polynomial does not have a single greatest common factor (GCF) for all terms. One effective method for factoring polynomials with four terms is called factoring by grouping. This technique involves grouping the polynomial’s terms into two pairs and then factoring out the GCF from each pair.

For example, consider the polynomial $x^3 + 2x^2 + 3x + 6$. First, group the terms into two pairs: $(x^3 + 2x^2)$ and $(3x + 6)$. The GCF of the first pair is $x^2$, and factoring it out leaves $(x + 2)$. The GCF of the second pair is \$3$, and factoring it out also leaves $(x + 2)$. Since both groups share the common binomial factor $(x + 2)$, you can factor this out, resulting in the expression $(x + 2)(x^2 + 3)\(.

It is important to note that sometimes the initial grouping of the first two and last two terms may not yield useful GCFs. In such cases, rearranging the terms to create pairs with common factors is necessary. This flexibility ensures that factoring by grouping can be applied effectively to a variety of four-term polynomials.

After factoring, always verify your result by multiplying the factors back together using the FOIL method (First, Outer, Inner, Last). For the example above, multiplying \)(x + 2)(x^2 + 3)$ gives $x^3 + 2x^2 + 3x + 6$, confirming the factorization is correct.

Mastering factoring by grouping enhances your ability to simplify polynomials and solve polynomial equations, which are essential skills in algebra and higher-level mathematics.

Problem

Use grouping to factor out the polynomial.
$x y + 2 x + 3 y + 6$

$(x + 3) (y + 2)$

$(x + 2) (y + 3)$

$(x + 3) (y + 6)$

$(x + 2) (y + 6)$

Problem

Use grouping to factor out the polynomial.
$2 a b + 4 a + 3 b^{2} + 6 b$

$(2 a + 3) (b + 2)$

$(2 a + b) (b + 6)$

$(b + 2) (2 a + 3 b)$

$(a + 3 b) (2 b + 2)$

example

Factoring Polynomials by Grouping Example 4

Video duration:

Factoring Polynomials by Grouping Example 4 Video Summary

When factoring polynomials using the grouping method, it is important to recognize that the terms may need to be rearranged to identify common factors effectively. Sometimes, the first two terms and the last two terms do not share a greatest common factor (GCF) initially, so rearranging the terms can reveal a more straightforward path to factorization.

For example, consider a polynomial where the terms ab, b, and -6 do not have a common factor, nor do the terms -2a and 3b. By swapping the second and last terms, the polynomial becomes ab + 3b - 2a - 6. Grouping the first two terms and the last two terms separately, the GCF of the first group (ab + 3b) is b, and factoring it out leaves a + 3. The GCF of the second group (-2a - 6) is -2, and factoring it out also leaves a + 3. Since both groups share the common binomial factor a + 3, it can be factored out, resulting in the product of two binomials: (a + 3)(b - 2).

In another case, when factoring a polynomial like y² + 26y + 6y + 156, swapping the middle two terms to y² + 6y + 26y + 156 can simplify the process. Grouping the first two terms and the last two terms, the GCF of the first group (y² + 6y) is y, leaving y + 6 after factoring. The GCF of the second group (26y + 156) is 26, and factoring it out also leaves y + 6. Factoring out the common binomial y + 6 results in the factored form (y + 6)(y + 26).

This approach highlights the importance of flexibility in factoring by grouping, where rearranging terms can reveal common factors that are not immediately obvious. Recognizing and factoring out the greatest common factor from grouped terms, followed by factoring out the common binomial, allows for efficient simplification of polynomials into products of binomials.

example

Factoring Polynomials by Grouping Example 5

Video duration:

Factoring Polynomials by Grouping Example 5 Video Summary

When factoring a polynomial completely, it is essential to first identify and factor out the greatest common factor (GCF) to simplify the expression before applying other factoring techniques. Consider a polynomial with multiple terms; breaking down each term into its prime factors helps reveal the GCF. For example, if the terms include factors such as 2, 3, x, and y in various combinations, the common factor present in all terms can be identified by comparing these prime factorizations.

Once the GCF is determined, factor it out from each term. This process reduces the polynomial to a simpler form, making it easier to apply methods like factoring by grouping. Factoring by grouping involves dividing the polynomial into pairs or groups of terms that share common factors. Extracting these common factors from each group often reveals a common binomial factor, which can then be factored out to complete the factorization.

For instance, after factoring out the GCF of 3, the polynomial might simplify to an expression like \$3(2xy^2 + 4xy^2 + 3x + 6y)$. Grouping the terms as $(2xy^2 + 4xy^2)$ and $(3x + 6y)$ allows factoring out $2xy$ from the first group and $3$ from the second, resulting in $(x + 2)$ as a common binomial factor. Factoring this binomial out leads to the fully factored form:

\[3(x + 2)(2xy + 3)\]

This approach highlights the importance of systematically breaking down polynomials by first extracting the greatest common factor and then using grouping techniques to identify and factor common binomial expressions. Mastery of these steps enhances problem-solving efficiency and deepens understanding of polynomial structures.

Here’s what students ask on this topic:

To find the greatest common factor (GCF) of polynomial terms, first write each term in its prime factorization form, including both numerical coefficients and variables with exponents. Identify the common prime factors shared by all terms. Multiply these common factors together to get the GCF. For example, for the terms $t^{2}$ and $t$ with coefficients 9 and 54, factor 9 as $3 × 3$ and 54 as $3 × 3 × 2 × 3$ . The common prime factors are two 3's and one $t$ , so the GCF is $9 t$ . This GCF can then be factored out of the polynomial to simplify it.

Factoring a polynomial using the greatest common factor (GCF) involves three main steps. First, identify the GCF by finding the largest factor that divides all terms, including variables and coefficients. Second, rewrite each term as a product of the GCF and another factor. For example, if the GCF is $6 x$ and a term is $12 x^{3}$ , rewrite it as $6 x × 2 x^{2}$ . Third, factor out the GCF from the polynomial, leaving a simpler polynomial inside the parentheses. Finally, check your work by redistributing the GCF to ensure you get the original polynomial.

Factoring by grouping is useful when a polynomial has four terms. The process involves grouping the terms into two pairs, factoring out the greatest common factor (GCF) from each pair, and then factoring out the common binomial factor from the resulting expression. For example, in the polynomial $x^{3} + 2 x^{2} + 3 x + 6$ , group the first two terms and the last two terms. Factor out $x^{2}$ from the first group and 3 from the second, resulting in $x^{2} (x + 2) + 3(x + 2)$ . Then factor out the common binomial $(x + 2)$ to get the factored form. If the initial grouping does not yield common factors, rearranging terms may help.

To check your factoring work, multiply the factors back together using the distributive property or FOIL method for binomials. For example, if you factored a polynomial as $GCF × (expression)$ , multiply the GCF by each term inside the parentheses. The result should match the original polynomial exactly. This verification confirms that the factoring was done correctly and helps reinforce understanding of polynomial structure and factoring techniques.

Factoring out the greatest common factor (GCF) first simplifies the polynomial, making it easier to apply other factoring methods. Removing the GCF reduces coefficients and variable exponents, which can reveal simpler patterns or structures in the polynomial. This step often makes subsequent factoring techniques, such as factoring by grouping or special products, more straightforward and efficient. It is a foundational skill that helps prevent errors and streamlines the factoring process.

Your Beginning & Intermediate Algebra tutors

Factoring by Greatest Common Factor & Grouping: Videos & Practice Problems

GCF of Terms

GCF of Terms Video Summary

What is the greatest common factor of the following lists of numbers?
$42 comma 70 comma 98$

What is the greatest common factor of the following lists of numbers?
$19 comma 57$

GCF of Terms Example 1

GCF of Terms Example 1 Video Summary

What is the greatest common factor of the following sets of monomials?
$6 x comma 9 x squared comma 15 x cubed$

What is the greatest common factor of the following sets of monomials?
$30 m^{3} n^{2}, 45 m^{2} n^{4}, 75 m^{4} n^{3}$

Factoring the GCF out of Polynomials

Factoring the GCF out of Polynomials Video Summary

Factor the GCF out of:
$8 x plus 12$

Factoring the GCF out of Polynomials Example 2

Factoring the GCF out of Polynomials Example 2 Video Summary

Factor the GCF out of:
$6 x squared plus 9 x$

Factor the GCF from the polynomial.
$18 x^{3} y^{2} - 27 x^{2} y^{3} + 9 x^{4} y$

Factoring the GCF out of Polynomials Example 3

Factoring the GCF out of Polynomials Example 3 Video Summary

Factoring Polynomials by Grouping

Factoring Polynomials by Grouping Video Summary

Use grouping to factor out the polynomial.
$x y + 2 x + 3 y + 6$

Use grouping to factor out the polynomial.
$2 a b + 4 a + 3 b^{2} + 6 b$

Factoring Polynomials by Grouping Example 4

Factoring Polynomials by Grouping Example 4 Video Summary

Factoring Polynomials by Grouping Example 5

Factoring Polynomials by Grouping Example 5 Video Summary

Here’s what students ask on this topic:

How do you find the greatest common factor (GCF) of polynomial terms?

What is the process for factoring a polynomial using the greatest common factor?

When should you use factoring by grouping for polynomials?

How do you check your factoring work after factoring polynomials?

What is the importance of factoring out the greatest common factor before using other factoring methods?

Your Beginning & Intermediate Algebra tutors

Factoring by Greatest Common Factor & Grouping: Videos & Practice Problems

GCF of Terms

GCF of Terms Video Summary

What is the greatest common factor of the following lists of numbers?42,70,9842,70,98

What is the greatest common factor of the following lists of numbers?19,5719,57

GCF of Terms Example 1

GCF of Terms Example 1 Video Summary

What is the greatest common factor of the following sets of monomials?6x,9x2,15x36x,9x^2,15x^3

What is the greatest common factor of the following sets of monomials?30m3n2,45m2n4,75m4n330m^3 n^2, 45m^2 n^4,75m^4 n^3

Factoring the GCF out of Polynomials

Factoring the GCF out of Polynomials Video Summary

Factor the GCF out of:8x+128x+12

Factoring the GCF out of Polynomials Example 2

Factoring the GCF out of Polynomials Example 2 Video Summary

Factor the GCF out of:6x2+9x6x^2+9x

Factor the GCF from the polynomial.18x3y2−27x2y3+9x4y18x^3y^2-27x^2y^3+9x^4y

Factoring the GCF out of Polynomials Example 3

Factoring the GCF out of Polynomials Example 3 Video Summary

Factoring Polynomials by Grouping

Factoring Polynomials by Grouping Video Summary

Use grouping to factor out the polynomial.xy+2x+3y+6xy+2x+3y+6

Use grouping to factor out the polynomial.2ab+4a+3b2+6b2ab+4a+3b^2+6b

Factoring Polynomials by Grouping Example 4

Factoring Polynomials by Grouping Example 4 Video Summary

Factoring Polynomials by Grouping Example 5

Factoring Polynomials by Grouping Example 5 Video Summary

Here’s what students ask on this topic:

How do you find the greatest common factor (GCF) of polynomial terms?

How do you find the greatest common factor (GCF) of polynomial terms?

What is the process for factoring a polynomial using the greatest common factor?

What is the process for factoring a polynomial using the greatest common factor?

When should you use factoring by grouping for polynomials?

When should you use factoring by grouping for polynomials?

How do you check your factoring work after factoring polynomials?

How do you check your factoring work after factoring polynomials?

What is the importance of factoring out the greatest common factor before using other factoring methods?

What is the importance of factoring out the greatest common factor before using other factoring methods?

Your Beginning & Intermediate Algebra tutors

What is the greatest common factor of the following lists of numbers?
$42 comma 70 comma 98$

What is the greatest common factor of the following lists of numbers?
$19 comma 57$

What is the greatest common factor of the following sets of monomials?
$6 x comma 9 x squared comma 15 x cubed$

What is the greatest common factor of the following sets of monomials?
$30 m^{3} n^{2}, 45 m^{2} n^{4}, 75 m^{4} n^{3}$

Factor the GCF out of:
$8 x plus 12$

Factor the GCF out of:
$6 x squared plus 9 x$

Factor the GCF from the polynomial.
$18 x^{3} y^{2} - 27 x^{2} y^{3} + 9 x^{4} y$

Use grouping to factor out the polynomial.
$x y + 2 x + 3 y + 6$

Use grouping to factor out the polynomial.
$2 a b + 4 a + 3 b^{2} + 6 b$