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

Factoring polynomials involves identifying the greatest common factor (GCF) by analyzing the prime factorization of terms, which may include variables with exponents. For example, factoring $\mathrm{9t^2}+\mathrm{54t}$ involves extracting the GCF $\mathrm{9t}$. When polynomials have four terms, factoring by grouping pairs and factoring out common binomials simplifies expressions, such as $\mathrm{x^3}+\mathrm{2x^2}+\mathrm{3x}+6$. Checking work with FOIL confirms accuracy. These methods enhance understanding of terms, coefficients, and polynomial structure.

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 3 × 3 × 3 × 2 × 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(𝑡 + 6).

This process can be generalized: first, identify the GCF by prime factorization; second, rewrite each term as the product of the GCF and a remaining factor; third, factor out the GCF to simplify the polynomial. To verify the factorization, multiply the GCF back through the terms to ensure the original polynomial is recovered.

Consider a more complex polynomial: 6𝑥 + 12𝑥³ − 24𝑥⁴. The prime factorization of each term is:

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

The common factors are 2, 3, and one 𝑥, so the GCF is 6𝑥. Rewriting each term:

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

Factoring out 6𝑥 yields: 6𝑥(1 + 2𝑥² − 4𝑥³). This simplified form is easier to work with and can be checked by redistributing 6𝑥.

Recognizing when a GCF exists is straightforward when all terms share common numerical factors or variables. Factoring out the GCF is often the first and simplest step in polynomial factorization, helping to reduce complexity before applying other methods. Mastery of this technique enhances problem-solving efficiency and lays the groundwork for more advanced factoring strategies.

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 be simplified by identifying the greatest common factor (GCF) through prime factorization. To factor a polynomial such as 10a³b + 15a²b² + 25ab³, start by breaking down each term into its prime factors. For example, the first term factors into 2 × 5 × a × a × a × b, the second into 3 × 5 × a × a × b × b, and the third into 5 × 5 × a × b × b × b.

Next, determine the common factors present in all terms. Each term contains at least one 5, one a, and one b, so the GCF is 5ab. Rewriting each term as a product of the GCF and another factor helps in factoring the polynomial. For instance, 10a³b can be expressed as 5ab × 2a², 15a²b² as 5ab × 3ab, and 25ab³ as 5ab × 5b².

Factoring out the GCF from the entire polynomial yields 5ab(2a² + 3ab + 5b²). This process highlights the importance of prime factorization and recognizing common factors when simplifying polynomials with multiple variables. Understanding how to identify and extract the greatest common factor is essential for efficient polynomial factoring and 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 unfamiliar, it is helpful to apply the greatest common factor (GCF) method. Begin by identifying the factors of each term in the polynomial. For example, if the first term factors into 3, x, and (x + 4), and the second term factors into 5 and (x + 4), the common factor between both terms is (x + 4). Recognizing this shared factor allows you to factor it out of the entire polynomial. After factoring out (x + 4), the remaining expression inside the parentheses is the combination of the leftover factors from each term, which in this case is (3x - 5). Thus, the polynomial is factored as \((x + 4)(3x - 5)\), demonstrating how the greatest common factor simplifies the expression effectively.

Factoring Polynomials by Grouping

Factoring polynomials is a fundamental skill in algebra, especially when the polynomial has four terms. When a greatest common factor (GCF) is not apparent across all terms, factoring by grouping becomes a powerful method. This technique involves dividing the polynomial into two pairs of terms, factoring out the GCF from each pair, and then factoring out the common binomial factor that emerges.

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)\(. Since both groups share the common binomial \)(x + 2)\(, this can be factored out, resulting in the expression \)(x + 2)(x^2 + 3)\(.

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 wide range of four-term polynomials.

To verify the factorization, multiply the factors back together using the FOIL method: multiply the first terms, outer terms, inner terms, and last terms, then combine like terms. For the example above, multiplying \)(x + 2)(x^2 + 3)\( yields \)x^3 + 2x^2 + 3x + 6$, confirming the correctness of the factorization.

Mastering factoring by grouping enhances problem-solving skills in algebra and prepares students for more complex polynomial manipulations. Practicing this method builds confidence in recognizing patterns and applying strategic rearrangements to simplify expressions efficiently.

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. In such cases, swapping terms can reveal a more straightforward path to factorization.

For example, consider a polynomial where the original grouping does not yield common factors. By rearranging the terms, such as switching the second and last terms, the polynomial can be grouped into pairs that each have a GCF. After grouping, factor out the GCF from each pair. If both groups share a common binomial factor, this binomial can be factored out, resulting in the product of two binomials. This process highlights the importance of flexibility in term arrangement to facilitate factoring by grouping.

In another case, even when the initial grouping has common factors, swapping the middle terms can still lead to the same factored form. Group the terms into two pairs, factor out the GCF from each pair, and then factor out the common binomial. This method ensures that the polynomial is expressed as the product of two binomials, simplifying the expression.

Overall, factoring by grouping relies on identifying the greatest common factor within pairs of terms and recognizing common binomial factors. Rearranging terms can be a strategic step to uncover these factors, making the factoring process more efficient and systematic.

Factoring Polynomials by Grouping Example 4

To factor a polynomial completely, it is effective to start by identifying and factoring out the greatest common factor (GCF). This initial step simplifies the polynomial, making subsequent factoring methods easier to apply. For example, consider a polynomial with terms that can be broken down into their prime factors. By expressing each term as a product of prime numbers and variables, you can determine the common factors shared by all terms.

Suppose the terms break down as follows: the first term includes factors 2, 3, x, x, and y; the second term includes 2, 2, 3, x, y, and y; the third term includes 3, 3, and x; and the fourth term includes 2, 3, 3, and y. Observing these, the number 3 appears in every term, indicating that 3 is the greatest common factor.

Factoring out 3 from each term, the polynomial can be rewritten as:

\$3(2x^2y + 4xy^2 + 3x + 6y)\(

Next, apply the factor by grouping method to the simplified polynomial inside the parentheses. Group the first two terms and the last two terms separately:

Group 1: \)2x^2y + 4xy^2\(

Group 2: \)3x + 6y\(

Within the first group, the common factors are 2, x, and y, so factor out \)2xy\(:

\)2xy(x + 2)\(

Within the second group, the common factor is 3, so factor out 3:

\)3(x + 2)\(

Now, both groups contain the common binomial factor \)(x + 2)\(, which can be factored out:

\)3(x + 2)(2xy + 3)$

This expression represents the polynomial fully factored. The process highlights the importance of first extracting the greatest common factor to simplify the polynomial, then using grouping to identify common binomial factors, ultimately leading to complete factorization.

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) of a polynomial, first write out the prime factorization of each term, including variables with their exponents. Identify the common prime factors and variables that appear in every term. The GCF is the product of these common factors with the smallest exponents. For example, for the terms $t^2$ and $t$, the common variable factor is $t$. If the coefficients share factors, include those as well. Once the GCF is identified, factor it out by rewriting each term as a product of the GCF and another factor, then express the polynomial as the GCF times the sum of these factors. This simplifies the polynomial and prepares it for further factoring if needed.

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 expressions share a common binomial factor, factor that out next. For example, given $x{x}^3+\; 2x^2+\; 3x+\; 6$, group as ($x{x}^3+\; 2x^2$) and ($3x+\; 6$). 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 $(x\; +\; 2)(x^2+\; 3)$. This method simplifies complex polynomials by breaking them into manageable parts.

How can you check your work after factoring a polynomial?

After factoring a polynomial, you can check your work by redistributing or expanding the factors to see if you get the original polynomial. For polynomials factored by the greatest common factor (GCF), multiply the GCF back by the remaining terms. For example, if you factored $9t^2\; -\; 54t$ as $9t(t\; -\; 6)$, multiply $9t$ by each term inside the parentheses to get $9t^2\; -\; 54t$. For polynomials factored by grouping, use the FOIL method to multiply the binomials. For instance, if you factored $x^3\; +\; 2x^2\; +\; 3x\; +\; 6$ as $(x\; +\; 2)(x^2\; +\; 3)$, multiply each term in the first binomial by each term in the second and combine like terms. If the result matches the original polynomial, your factoring is correct.

When should you use factoring by grouping instead of factoring by the greatest common factor?

Factoring by grouping is typically used when a polynomial has four terms and factoring out a single greatest common factor (GCF) from all terms is not possible. If the polynomial does not have a GCF common to all terms, but can be split into pairs where each pair has its own GCF, grouping is effective. You group the terms into two pairs, factor out the GCF from each pair, and then factor out the common binomial factor. If the polynomial has fewer than four terms or all terms share a GCF, factoring by the GCF is usually the best first step. Always try factoring out the GCF first to simplify the polynomial before considering grouping.

How do you factor out the greatest common factor from terms with variables and exponents?

To factor out the greatest common factor (GCF) from terms with variables and exponents, identify the lowest power of each variable that appears in all terms. For example, if the terms are $6x$, $12x^3$, and $24x^4$, the variable $x$ appears in all terms with exponents 1, 3, and 4 respectively. The smallest exponent is 1, so $x$ to the first power is part of the GCF. Next, find the GCF of the coefficients (6, 12, 24), which is 6. Therefore, the GCF is $6x$. Factor this out by rewriting each term as the product of $6x$ and another factor, then express the polynomial as $6x(1\; +\; 2x^2\; -\; 4x^3)$. This method ensures you factor out the largest common factor including variables with exponents.