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. These methods simplify polynomials, aiding in solving equations and understanding terms, coefficients, exponents, and polynomial degrees, essential for mastering multivariable polynomials and applying the FOIL technique for verification.

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. Before tackling polynomial factoring, it is essential to understand the concept of the greatest common factor (GCF), which plays a crucial role in simplifying terms and expressions.

The greatest common factor of a set of numbers or terms is the product of all their common prime factors. To find the GCF, start by expressing each number or term in its prime factorization form. For example, the number 9 factors into $3 \times 3$, while 54 factors into $3 \times 3 \times 2 \times 3$. Identifying the common prime factors between 9 and 54 reveals two 3s, so the GCF is $3 \times 3 = 9$.

This method extends to algebraic terms as well. Consider the terms \$9t^2$ and $54t$. Factoring these gives \(9t^2 = 3 \times 3 \times t \times t$ and $54t = 3 \times 3 \times 2 \times 3 \times t$. The common prime factors here are two 3s and one \)t$, so the GCF is \(3 \times 3 \times t = 9t$.

For more complex examples, such as \)12a^2$, $30a^3$, and $42a^5$, break down each term into prime factors: \(12a^2 = 2 \times 2 \times 3 \times a \times a$, $30a^3 = 3 \times 2 \times 5 \times a \times a \times a$, and $42a^5 = 2 \times 3 \times 7 \times a \times a \times a \times a \times a$. The common factors across all terms are \)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. This perspective is especially useful when factoring polynomials, as it helps identify the greatest factor to factor out from all terms, simplifying the expression and making further factoring steps more manageable.

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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 a²b, ab³, and a³b², start by expressing each term as a product of its prime factors. The first term, a²b, can be written as a × a × b. The second term, ab³, breaks down into a × b × b × b. The third term, a³b², is a × a × a × b × b.

To determine the GCF, identify the common factors present in all three monomials. Each term contains at least one a and one 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?
$6 x comma 9 x squared comma 15 x cubed$

$x$

$3x$

$15x$

$x^3$

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

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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 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.

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

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Factoring the GCF out of Polynomials Example 2 Video Summary

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.

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 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.

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 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.

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

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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. 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.

example

Factoring Polynomials by Grouping Example 5

Video duration:

Factoring Polynomials by Grouping Example 5 Video Summary

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:

To find the greatest common factor (GCF) of polynomial terms, first write each term in its prime factorization form, including variables with exponents. For example, for terms like $t^{2}$ and $t$ , express them as $t × t$ and $t$ respectively. Then, identify all common prime factors and variables with the smallest exponents shared by all terms. Multiply these common factors together to get the GCF. For instance, for $9t^2$ and $54t$ , the GCF is $9t$ because both share factors of $3 × 3 × t$ . This process helps simplify polynomials by factoring out the GCF before applying other factoring methods.

To factor a polynomial using the greatest common factor (GCF), follow these steps: First, identify the GCF by finding the largest factor that divides all terms, including variables with the lowest exponents. Second, rewrite each term as a product of the GCF and another factor. For example, if the GCF is $6x$ and a term is $12x^3$ , rewrite it as $6x × 2x^2$ . Third, factor out the GCF from the polynomial, leaving a simpler polynomial inside parentheses. For example, $6x + 12x^3 - 24x^4$ becomes $6x(1 + 2x^2 - 4x^3)$ . Finally, check your work by redistributing the GCF to ensure you get the original polynomial.

Factoring by grouping is 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, in the polynomial $x^3 + 2x^2 + 3x + 6$ , group as $(x^3 + 2x^2)$ and $(3x + 6)$ . Factor out the GCF from each group: $x^2$ from the first and $3$ from the second, resulting in $x^2(x + 2) + 3(x + 2)$ . Since both groups share the binomial $(x + 2)$ , factor it out to get $(x + 2)(x^2 + 3)$ . If the initial grouping doesn't work, try rearranging terms to find pairs with common factors.

To check if your factoring is correct, use redistribution or the FOIL method. For factoring by GCF, multiply the factored GCF by the remaining polynomial inside the parentheses and verify if it equals the original polynomial. For example, if you factored $9t^2 + 54t$ as $9t(t + 6)$ , multiply back: $9t × t = 9t^2$ and $9t × 6 = 54t$ . For factoring by grouping, multiply the binomials using FOIL (First, Outer, Inner, Last) and confirm the product matches the original polynomial. This step ensures accuracy and helps catch mistakes.

Prime factorization is crucial in factoring polynomials because it helps identify the greatest common factor (GCF) among terms. By breaking down coefficients into their prime factors and expressing variables with exponents as repeated factors, you can clearly see which factors are common across all terms. This allows you to extract the largest possible GCF, simplifying the polynomial effectively. For example, factoring $54$ into $2 × 3 × 3 × 3$ helps find common factors with other terms. Understanding prime factorization also aids in recognizing when to apply other factoring methods like grouping.

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