Factoring out the GCF 04:53 minutes

Video Transcript

Transcript Factoring out the GCF

Meet Gi Na. She loves to play her flute outside in the fresh air, close to nature. Gi Na's friend, Chaika, loves to do Ollies on her skateboard at the city skate park. Whoa, look at her go! This is Fernando. He likes to hang out in his room with his Chameleon, Oscar. These three friends have three VERY different interests.

GCF in everyday life

But, Gi Na likes to do lots of other things too. She likes crossword puzzles, glass blowing, painting, origami, karaoke, jewelry making, do-it-yourself projects, Sudoku, sewing, knitting. It’s a long list, you get the picture. In addition to skateboarding, Chaika likes photography, music, snowboarding, snake boarding, watching movies and tv, metal working, graffiti, hip hop and karaoke. Another long list of interests.

Fernando also likes drawing, acting, doing Sudoku puzzles, collecting insects, roasting coffee, arranging flowers, singing Karaoke, cheerleading, and making movies. The three friends want to do something together, but something they ALL enjoy, an interest that’s common to all three. How can we figure this out? It's like finding the Greatest Common Factor, or GCF for short.

The Greatest Common Factor in math

We can relate this real world problem to a math topic, finding the Greatest Common Factor. To make polynomials easier to work with, we factor out the Greatest Common Factor and to do this, we undo the Distributive Property. You can also think of it as the reverse of the Distributive Property.

First example

Let’s take a look at an example. 4x + 28. 4 is common to both terms, so by undoing or reversing the Distributive Property, we factor out the 4. Notice how I write this down, outside parentheses is the GCF, and inside parentheses is what's left of the expression after dividing all terms by the GCF.

Second example

Let’s try a second example, 4x² + 28x. You can write this as (4)(x)(x) + (4)(7)(x) Hmm, both terms have a common factor of 4 and they both have an 'x' so we can factor out 4 and also an 'x' and in a reverse of the Distributive Property, we display the GCF, 4x, outside the parentheses and what's left of the expression inside by dividing all terms by the GCF.

Last check

Don't forget: to check your work, you can apply the Distributive Property. Looking good! Sometimes math teachers like to assign problems that look impossible to solve, but if you know the right steps, they're not that bad. Let’s try one of those now.

Oh boy, it’s a doozey, 10x³y² + 18x²y - 4x. (2 * 5) (x * x * x) (y * y) + (2 * 3 * 3) (x * x) * y - (2 * 2) (x) 2x is the GCF. So let's factor it out. Okay, what's the result 2x times the trinomial 5x²y² + 9xy + 2. Wow, that was a lot of work! Do you need to completely factor every term, everytime? No, probably most of the time, you can figure it out without writing out all the factors.

Summary

Okay, let's summarize. To find the Greatest Common Factor of a polynomial, find the factors common to each term in the polynomial. Back to our three friends. Did they figure out something they all like to do? Li Na and Leon like Sudoko, but Ayumi is not a fan, so that's not the GCF or the common interest for all three. But… they all like Karaoke! It’s their GCF! But sadly, their taste in music is not a GCF.

Factoring out the GCF Übung

Du möchtest dein gelerntes Wissen anwenden? Mit den Aufgaben zum Video Factoring out the GCF kannst du es wiederholen und üben.

  • Describe the greatest common factor (GCF).

    Tipps

    To determine the GCF, let's look at an example:

    $6 = 2 \times 3$ and $8 = 2 \times 2 \times 2$

    Both numbers, $6$ and $8$, have the factor $2$ in common, so $2$ is the GCF.

    Now let's take a look at another example:

    $12 = 2 \times 2 \times 3$ and $8 = 2 \times 2 \times 2$

    As you can see, both numbers have the factor 2 listed twice. So the GCF is $2 \times 2 = 4$.

    Lösung

    Wow, Gi Na, Chiako, and Fernando have a lot of hobbies. They decide to spend time together and want to do an activity they all enjoy - a common interest.

    The common interest in this example corresponds to the Greatest Common Factor - GCF.

    Let's have a look at the list to the right:

    • Gi Na and Fernando love Sudoku puzzles, but Chiako doesn't care for them.
    • However, we can see that everyone likes karaoke.
    So karaoke is the common interest of the three friends

    Mathematically the GCF is defined as follows:

    To find the greatest common factor of a polynomial, find the factors common to each term in the polynomial.

  • Determine the greatest common factor for all the terms listed in an expression.

    Tipps

    Remember, you are looking for the greatest common factor, not just any factor.

    Make a list of factors for each term and see which factors are common to all terms.

    The product of the factors common to each term is called the greatest common factor.

    Lösung

    First, we factor each term and then look for the common factors:

    $4x^2 + 28x = (2 \times 2) \times (x \times x)+ (2 \times 2 \times 7) \times (x)$

    Now we see that two $2$s and $x$ are factors common to all terms. So the product of $2 \times 2 \times x$ is the greatest common factor, $4x$.

    So when we factor out the GCF from the original expression, we get:

    $4x^2 + 28x = 4x(x + 7)$

    Again, you can check your solution using the Distributive Property. Multiply $4x$ by each term listed inside the parentheses.

  • Determine the greatest common factor.

    Tipps

    You can check the results by using the Distributive Property.

    Look at the following example:

    • $6x^2y=(2 \times 3)(x \times x)(y)$
    • $8xy^2 = (2 \times 2 \times 2)(x)(y \times y)$
    As you can see, both terms have the factors $2$, $x$, and $y$ in common.

    Therefore, the GCF is $2xy$.

    To factor a polynomial, write the polynomial as a product where one factor is the GCF, and the other factor is a polynomial listing all the remaining terms.

    For example:

    $6x^2y - 8xy^2 = (2 \times 3)(x \times x)(y) - (2 \times 2\times 2)(x)(y \times y) = 2xy(3x - 4y)$.

    As you can see, both terms have the factors $2$, $x$, and $y$ in common. So the greatest common factor of both terms is $2xy$.

    Factoring out $2xy$ from $6x^2y$ leaves $3x$. And factoring out $2xy$ from $-8xy^2$ leaves $-4y$. Therefore, the remaining terms comprises the polynomial $3x - 4y$.

    Lösung

    $10x^3y^2 + 18x^2y - 4x$

    Oh boy, what a funky polynomial.

    To factor out the GCF:

    We write each term as a product:

    • $10x^3y^2 = (2 \times 5)( x \times x \times x)(y \times y)$
    • $18x^2y = (2 \times 3 \times 3)(x \times x)(y)$
    • $4x = (2 \times 2)(x)$
    You can see, that $2$ and $x$ are common factors. So $2x$ is the greatest common factor.

    Now we factor out the GCF to get:

    $10x^3y^2 + 18x^2y - 4x = 2x(5x^2y^2 + 9xy - 2)$

    You can check your solution using the Distributive Property. To do this, multiply $2x$ by each term listed inside the parentheses.

  • Name the Greatest Common Factor for each given term.

    Tipps

    To find the greatest common factor, first determine all the common factors, and then calculate the greatest common factor as the product of the common factors.

    For example:

    Find the GCF of $32xy$ and $36x^2$. First factor both terms to primes:

    • $32xy = 4 \times 8 \times x \times y$
    • $36x^2 = 4 \times 9 \times x \times x$
    As you can see, both terms have the factors $4$ and $x$ in common, so $4 \times x = 4x$ is the greatest common factor.

    Lösung

    We can write each number as the product of its prime factors.

    For example, the prime factors of $32x^2$ are $2 \times 2 \times 2 \times 2 \times 2 \times x \times x$.

    (1) Find the GCF for the expression $24x^3-16x^2+8x$.

    First, prime factor each term:

    • $24x^3 = 2 \times 2 \times 2 \times 3 \times x \times x \times x$
    • $16x^2 = 2 \times 2 \times 2 \times 2 \times x \times x$
    • $8x = 2 \times 2 \times 2 \times x$
    We can see that the three terms have three $2$s as well as an $x$ in common,so the product of these common factors: $2 \times 2 \times 2 \times x=8x$ is the GCF. We can factor this out of our original expression as follows:

    $24x^3 - 16x^2 + 8x = 8x(3x^2 - 2x + 1)$

    $~$

    (2) $12x^2+6x-18$

    Factoring each term, we get:

    • $12x^2 = 2 \times 2 \times 3 \times x \times x$
    • $6x = 2 \times 3 \times x$
    • $18 = 2 \times 3 \times 3$
    Here, we have only one common factor, $6$. So this is also our GCF. When we factor $6$ from our original expression, we write:

    $12x^2 + 6x - 18 = 6(2x^2 + x - 3)$

    $~$

    (3) $36x^4 + 18x^2 - 24xy$

    Factoring each term, we list the prime factors:

    • $36x^4 = 2 \times 2 \times 3 \times 3 \times x \times x \times x \times x$
    • $18x^2 = 2 \times 3 \times 3 \times x \times x$
    • $24xy = 2 \times 2 \times \times 2 \times 3 \times x \times y$
    The common factors of all three terms are $6$ and $x$, so the GCF $6x$. Factoring this out of the original expression, we can write:

    $36x^4 + 18x^2 - 24xy = 6x(6x^3 + 3x - 4y)$.

    You can always check your solution using the Distributive Property.

  • Find the greatest common factor.

    Tipps

    You don't have to determine the sum of the fruits.

    Determine the common factor for each type of fruit.

    Lösung

    First, consider how many of each fruit the friends brought for the camping trip.

    Apples

    • $4$ apples from Gi Na
    • $2$ apples from Chiako
    • $2$ apples from Fernando
    Since $2$ is a factor that Chiako and Fernando have in common, we now need to see if the number of apples Gi Na brought also has $2$ as a factor. $4 = 2 \times 2$, So $2$ is a factor of $4$.

    $2$ apples is our greatest common factor.

    Bananas

    • $2$ bananas from Gi Na
    • $4$ bananas from Chiako
    • $6$ bananas from Fernando
    • $2 = 2 \times 1$
    • $4 = 2 \times 2$
    • $6 = 2 \times 3$
    So $2$ bananas is our greatest common factor.

    Kiwis

    • $1$ kiwi from Gi Na
    • $2$ kiwis from Chiako
    • $0$ kiwis from Ferndando
    Since Fernando didn't bring any kiwis and since $0$ does not have any factors, there is no GCF for kiwis.

  • Consider each term and find the greatest common factor.

    Tipps

    First factor each number or term as much as possible.

    For example. to determine the GCF of $45$ and $60$, factor the numbers:

    • $45 = 3 \times 3\times 3\times 5$
    • $60 = 2 \times 2\times 3\times 5$
    Next check for factors that are common to both numbers. $45$ and $60$ have the factors $3$ and $5$ in common.

    For the final step, multiply the common factors to find the GCF:

    $3 \times 5 = 15$

    Lösung

    How can we determine the greatest common factor of two or more given terms?

    To find the greatest common factor of a polynomial, find the factors common to each term in the polynomial.

    If we find more than one factor common to each term, multiply those factors together:

    (1) GCF of $35$ and $77$:

    • $35 = 5 \times 7$
    • $77 = 7 \times 11$
    So the GCF of $35$ and $77$ is $7$.

    $~$

    (2) GCF of $36$ and $24$:

    • $36 = 2 \times 2\times 3\times 3$
    • $24 = 2 \times 2\times 2\times 3$
    The common factors are two $2$s and one $3$. Multiply these factors, and the product is the GCF $2 \times 2 \times 2 = 12$.

    $~$

    (3) GCF of $15x^2$ and $30x$:

    • $15x^2 = 3 \times 5 \times x \times x$
    • $30x = 2 \times 3 \times 5 \times x$
    So the common factors are $3$, $5$, and $x$, so the GCF is the product of these factors: $3 \times 5 \times x = 15x$.

    $~$

    (4) GCF of $15xy$ and $35x^2$:

    • $15xy = 3 \times 5 \times x \times y$
    • $35x^2 = 5 \times 7 \times x \times x$
    The common factors are $5$ and $x$. The GCF is $5 \times x = 5x$.