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 out the GCF from a multivariable polynomial simplifies it into a product of the GCF and a binomial or trinomial. When polynomials have four terms, factoring by grouping pairs and extracting common factors reveals shared binomials, enabling further factoring. Checking work with the FOIL method confirms accuracy. Understanding terms, coefficients, and exponents is essential for mastering polynomial factoring and simplifying expressions efficiently.

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 out the greatest common factor (GCF), which is the largest factor shared by all terms in a polynomial. To find the GCF, start by determining the prime factorization of each term. For example, consider the terms 9𝑡² and 54𝑡. The prime factors of 9𝑡² are 3 × 3 × 𝑡 × 𝑡, and for 54𝑡, they are 3 × 3 × 3 × 2 × 𝑡. The common factors are two 3s and one 𝑡, so the GCF is 9𝑡.

Once the GCF is identified, rewrite each term as a product involving the GCF. This process is like reverse distribution. For instance, 9𝑡² can be expressed as 9𝑡 × 𝑡, and 54𝑡 as 9𝑡 × 6. Factoring out the GCF then gives 9𝑡(𝑡 − 6). To verify, distribute 9𝑡 back through the parentheses to ensure the original polynomial is recovered.

Applying this method to a more complex polynomial such as 6𝑥 + 12𝑥³ − 24𝑥⁴ involves first finding the GCF by prime factorization. The factors of 6𝑥 are 2 × 3 × 𝑥; for 12𝑥³, they are 2 × 2 × 3 × 𝑥 × 𝑥 × 𝑥; and for 24𝑥⁴, they are 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.

Next, express each term as a product with 6𝑥: 6𝑥 = 6𝑥 × 1, 12𝑥³ = 6𝑥 × 2𝑥², and 24𝑥⁴ = 6𝑥 × 4𝑥³. Factoring out 6𝑥 yields 6𝑥(1 + 2𝑥² − 4𝑥³). This factored form simplifies the polynomial and makes further operations easier. Always confirm your factorization by redistributing the GCF to retrieve the original expression.

Recognizing when a GCF exists is straightforward if all terms share common numerical factors or variables. Factoring out the GCF is often the first step in simplifying polynomials before applying more advanced factoring techniques. Mastery of this method enhances problem-solving efficiency and deepens understanding of polynomial structures.

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

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

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 shared by 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 = 5ab × 2a², 15a²b² = 5ab × 3ab, and 25ab³ = 5ab × 5b².

Factoring out the GCF from the entire polynomial yields 5ab(2a² + 3ab + 5b²). This process highlights the importance of prime factorization and identifying the greatest common factor when simplifying polynomials with multiple variables. Understanding how to factor out the GCF not only simplifies expressions but also lays the foundation for more advanced algebraic techniques.

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

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

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 concept of the greatest common factor (GCF). 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 the product of the greatest common factor and the simplified binomial, resulting in the factored form: \[(x + 4)(3x - 5).\] This method reinforces the importance of identifying common factors to simplify polynomials efficiently.

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 for 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 a common factor 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 (First, Outer, Inner, Last). For the example above, multiplying \)(x + 2)(x^2 + 3)\( gives:

\[x \times x^2 = x^3\]

\[x \times 3 = 3x\]

\[2 \times x^2 = 2x^2\]

\[2 \times 3 = 6\]

Combining these terms results in the original polynomial \)x^3 + 2x^2 + 3x + 6$, confirming the correctness of the factorization.

Mastering factoring by grouping enhances problem-solving skills and prepares students for more complex polynomial manipulations. Practicing this method builds a strong foundation for algebraic proficiency and supports understanding of polynomial structure and factorization strategies.

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

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

Answer: (A)

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

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

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, rearranging the terms can reveal a suitable grouping for factoring.

For example, consider the polynomial ab + b - 2a - 6. Initially, the pairs ab and b do not have a common factor, nor do -2a and -6. However, by swapping the second term b with the last term -6, the polynomial becomes ab + 3b - 2a - 6. Grouping the first two terms and the last two terms separately, the GCF of ab + 3b is b, and factoring it out leaves a + 3. Similarly, the GCF of -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 factored form (a + 3)(b - 2).

In another example, the polynomial y² + 26y + 6y + 156 can be rearranged by swapping the middle terms to y² + 6y + 26y + 156. Grouping the first two terms and the last two terms, the GCF of y² + 6y is y, and factoring it out leaves y + 6. The GCF of 26y + 156 is 26, and factoring it out also leaves y + 6. Factoring out the common binomial y + 6 yields the final factored form (y + 6)(y + 26).

This approach highlights the importance of flexibility in factoring by grouping, emphasizing that rearranging terms can simplify the process by revealing common factors. Recognizing and factoring out the greatest common factor from grouped terms, followed by factoring out the common binomial, are key steps in efficiently factoring polynomials.

Factoring Polynomials by Grouping Example 4

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, rearranging the terms can reveal a more straightforward path to factorization.

For example, consider a polynomial where the original grouping does not yield common factors. By swapping terms, such as exchanging the second and last terms, the polynomial can be rewritten to facilitate factoring by grouping. After rearrangement, group the terms into two pairs. Identify the GCF of each pair: for instance, factoring out b from the first group and factoring out −2 from the second group. This process results in a common binomial factor, which can then be factored out, simplifying the polynomial into the product of two binomials.

In another case, even if the original grouping has common factors, swapping the middle terms can still lead to the same factored form. Group the terms accordingly, find the GCF for each group, and factor them out. When both groups share the same binomial factor, factor it out to express the polynomial as the product of two binomials.

This approach highlights the flexibility and strategic thinking involved in factoring by grouping. Recognizing when to rearrange terms and how to identify the greatest common factor in each group is essential for simplifying polynomials efficiently. The key is to create groups that share a common binomial factor, enabling the polynomial to be factored completely.

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 identifying the prime factorization of each term, including both numerical coefficients and variables with their exponents. For example, consider the polynomial $t^2+\; 54t$. First, factor each term: 9t² breaks down into 3 × 3 × t × t, and 54t breaks down into 3 × 3 × 3 × 2 × t. Then, identify the common factors in all terms. Here, both terms share two 3's and one t, so the GCF is 9t. Factoring out the GCF means rewriting each term as a product of the GCF and another factor, then expressing the polynomial as the GCF multiplied by the sum of these factors. This method simplifies the polynomial and prepares it for further factoring if needed.

What is the process of factoring polynomials by grouping?

Factoring by grouping is a method used when a polynomial has four terms. The process involves grouping the terms into two pairs and factoring out the greatest common factor (GCF) from each pair. For example, consider the polynomial $x^3+\; 2x2+\; 3x\; +\; 6$. Group the first two terms and the last two terms: ($x^3+\; 2x2$) and (3x + 6). Factor out the GCF from each group: $x2$ from the first group and 3 from the second, resulting in $x2(x\; +\; 2)\; +\; 3(x\; +\; 2)$. Since both terms share the binomial $(x\; +\; 2)$, factor it out to get $(x\; +\; 2)(x2+\; 3)$. This method simplifies the polynomial and is especially useful when direct factoring is not obvious.

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. For example, if you factored a polynomial as $(x\; +\; 2)(x2+\; 3)$, multiply the terms: First multiply $x\; \times \; x2=\; x3$, Outer multiply $x\; \times \; 3\; =\; 3x$, Inner multiply $2\; \times \; x2=\; 2x2$, and Last multiply $2\; \times \; 3\; =\; 6$. Adding these results gives $x3+\; 2x2+\; 3x\; +\; 6$, which should match the original polynomial. This confirms that the factoring is correct and helps reinforce understanding of polynomial structure.

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, making it easier to apply additional techniques like factoring by grouping or special products. For example, if all terms are multiples of 6x, factoring out 6x reduces the polynomial to a simpler form. This step can save time and reduce complexity, and it is often the first step in the factoring process. If no GCF exists, then other methods should be considered.

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

If the initial grouping of terms in factoring by grouping does not produce a greatest common factor (GCF) other than 1, you may need to rearrange the terms to find pairs that do share a GCF. For example, if grouping the first two and last two terms does not work, try grouping the first and third terms together and the second and fourth terms together. The goal is to create pairs where each has a GCF that can be factored out, leading to a common binomial factor. This flexibility in grouping is important to successfully factor polynomials with four terms.