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

Factoring polynomials involves identifying the greatest common factor (GCF) by analyzing the prime factorization of each term, including numerical coefficients and variables with exponents. For example, factoring $\mathrm{9t^2}+\mathrm{54t}$ yields a GCF of $\mathrm{9t}$. Another method, factoring by grouping, applies to four-term polynomials by pairing terms and factoring out common binomials. Checking work with the FOIL method ensures accuracy. These techniques simplify multivariable polynomials and prepare students to handle more complex expressions involving terms, coefficients, and exponents.

Factoring the GCF out of Polynomials

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.

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

• A. $2\left(4x+6\right)$
• B. $2\left(4x+3\right)$
• C. $4\left(2x+3\right)$
• D. $4\left(2x+6\right)$

Answer: (C)

Factoring the GCF out of Polynomials Example 1

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.

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

• A. $3x\left(2x+3\right)$
• B. $3\left(2x^2+3x\right)$
• C. $6x\left(x+3\right)$
• D. $3x\left(2x+6\right)$

Answer: (A)

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

• A. $9x^{2}y(2x^2-3xy+3x^2)$
• B. $9x^{2}y(2xy+3y^2+x^2)$
• C. $9x^{2}y(2x^2-3xy+x^3)$
• D. $9x^{2}y(2xy-3y^2+x^2)$

Answer: (D)

Factoring the GCF out of Polynomials Example 2

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:

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

Factoring Polynomials by Grouping

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.

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

• A. $\left(x+3\right)\left(y+2\right)$
• B. $\left(x+2\right)\left(y+3\right)$
• C. $\left(x+3\right)\left(y+6\right)$
• D. $\left(x+2\right)\left(y+6\right)$

Answer: (A)

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

• A. $\left(2a+3\right)\left(b+2\right)$
• B. $\left(2a+b\right)\left(b+6\right)$
• C. $\left(b+2\right)\left(2a+3b\right)$
• D. $\left(a+3b\right)\left(2b+2\right)$

Answer: (C)

Factoring Polynomials by Grouping Example 3

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.

Factoring Polynomials by Grouping Example 4

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:

How do you find the greatest common factor (GCF) when factoring polynomials?

To find the greatest common factor (GCF) when factoring polynomials, start by writing the prime factorization of each term, including both numerical coefficients and variables with their exponents. Identify the common prime factors shared by all terms. Multiply these common factors together to get the GCF. For example, for the polynomial $\mathrm{9t}{2}^{}+\mathrm{54t}$, the prime factors of $9t^2$ are $3\; imes\; 3\; imes\; t\; imes\; t$, and for $54t$ they are $2\; imes\; 3\; imes\; 3\; imes\; 3\; imes\; t$. The common factors are two 3's and one t, so the GCF is $9t$. Factoring out the GCF simplifies the polynomial and makes further factoring easier.

What is the process for factoring polynomials by grouping?

Factoring by grouping is used when a polynomial has four terms. First, group the terms into two pairs, usually the first two and the last two terms. Then, factor out the greatest common factor (GCF) from each pair. If the resulting binomials are the same, factor that binomial out. For example, for $x^3\; +\; 2x^2\; +\; 3x\; +\; 6$, group as $(x^3\; +\; 2x^2)$ and $(3x\; +\; 6)$. Factor out $x^2$ from the first group and $3$ from the second, giving $x^2(x\; +\; 2)\; +\; 3(x\; +\; 2)$. Since both terms contain $(x\; +\; 2)$, factor it out to get $(x\; +\; 2)(x^2\; +\; 3)$. This method simplifies polynomials that don't have a single GCF for all terms.

How can you check your factoring work using the FOIL method?

After factoring a polynomial, you can check your work by multiplying the factors back together using the FOIL method, which stands for First, Outer, Inner, Last. This method applies to multiplying two binomials. Multiply the first terms, then the outer terms, the inner terms, and finally the last terms, and then combine like terms. For example, if you factored $x^3\; +\; 2x^2\; +\; 3x\; +\; 6$ as $(x\; +\; 2)(x^2\; +\; 3)$, multiply using FOIL: $x\; imes\; x^2\; =\; x^3$, $x\; imes\; 3\; =\; 3x$, $2\; imes\; x^2\; =\; 2x^2$, and $2\; imes\; 3\; =\; 6$. Adding these gives the original polynomial, confirming the factoring is correct.

When should you try factoring out the greatest common factor before other methods?

It is always a good idea to first check if there is a greatest common factor (GCF) among all terms of a polynomial before applying other factoring methods. Factoring out the GCF simplifies the polynomial and can make subsequent factoring easier. For example, if all terms are even or share common variables, factor out the GCF first. This step reduces the polynomial to a simpler form, which might then be factored by grouping or other techniques. Skipping this step can make factoring more complicated than necessary.

What do you do if the first grouping in factoring by grouping does not produce a common factor?

If the initial grouping of terms in factoring by grouping does not yield a greatest common factor (GCF) other than 1, try rearranging the terms to create pairs that do share a GCF. For example, if grouping the first two and last two terms doesn't work, try grouping the first and third terms together and the second and fourth terms together. The goal is to find pairs where factoring out a GCF is possible. Once you find such pairs, factor out the GCF from each and proceed with factoring the common binomial factor.