WEBVTT
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In this video, we're going to discuss greatest common factors and factoring by grouping.
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In the product two times three which is equal to six, two and three are known as the factors
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of six. Two is a factor, and three is a factor of six. Similarly, in algebraic expressions,
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x plus two times x plus three is an expression written in factored form. When multiplied together,
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they give us the polynomial x squared plus five x plus six. So x plus two and x plus three
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are the factors of the polynomial, and again, this is known as factored form.
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The greatest common factor, or GCF, of a list of terms or monomials is the product of the
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GCF of the numerical coefficients and each GCF of the variable factors.
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Let's look at an example. I have two monomials here, twelve m n to the fifth and four m squared
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n to the seventh. To find the GCF of each of these two expressions, we begin by looking at the GCF
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of the numerical coefficients. The common factors would be the factors that these two coefficients
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both have. So let's write out the factors first of twelve. The factors of twelve are one, two,
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three, four, six, and twelve. Where the factors of four are one, two,
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and four. We can see that they have in common factors of one, two,
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and four. So the GCF, or greatest common factor, of the numerical coefficients is four. Now looking
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at the variable factors, we have m to the first power and m squared. m and m squared, which is
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the same as m times m. So the common factors here are m. They both share an m to the first power.
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For the next variable factor, we have n to the fifth and n to the seventh.
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n to the fifth can be written as n times itself five times, and n to the seventh
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can be written as n times itself seven times. Or another way it can be written
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as n to the fifth times n squared. So the greatest common factor of these is n to the fifth.
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Multiplying these GCF's together gives us four m n to the fifth. So the GCF is four m n to the fifth.
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In these next examples, we are asked to factor each polynomial by factoring out the GCF.
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Factoring a polynomial is the reverse process of using the distributive property to multiply
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factors together. So we'll begin with fourteen t plus twenty-one. We need to identify the GCF
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of the two terms, fourteen t and twenty-one. So recall to do that we look for the product
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of the GCF of the numerical coefficients, which would be fourteen and twenty-one, and we multiply
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that times the GCF of the variable factors. The GCF of fourteen and twenty-one is seven.
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And since there's only a t in this first term and not in the second, there is not a greatest common
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factor or common factor at all of the variable factor. So the GCF of the monomials here is seven.
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Using the reverse distributive property, we can rewrite this as seven open parentheses
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and then divide each of these two terms by the GCF. So fourteen t divided by seven
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leaves us with two t, and twenty-one divided by seven leaves us with three, so plus three. This
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is factored form of this polynomial. We can check our results by using the distributive property
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to distribute seven into the grouping symbols, and you would see that this expression is
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equivalent to fourteen t plus twenty-one. Now let's look at our next example.
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We have negative six x squared plus three x. Let's begin by finding the GCF of the
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two terms. We begin with the numerical coefficients, negative six and three.
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The GCF of these two numerical expressions is three. We can factor out either a positive or a negative numerical coefficient
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in any situation. The GCF of the variable factors, x squared and x, is x to the first power.
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So the GCF of the terms gives us three x. So let's factor out three x as our GCF.
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Using the reverse process of the distributive property means that we will divide each term
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by three x. So negative six x squared divided by three x leaves us with negative two x.
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And three x divided by three x leaves us with one. So we'll have a plus one. And again we
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can see that if we were to distribute three x through the grouping symbols,
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we would have the same polynomial that we began with, negative six x squared plus three x.
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Our next example is x to the fourth minus eight x to the third power. Let's begin by finding the GCF of these two terms.
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We begin by looking at the numerical coefficients one and eight. The greatest common factor of one
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and eight is just one. The greatest common factor of the variable factors x to the fourth and x to
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the third is x to the third. So the GCF is one x to the third or just x to the third. So we'll
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write this in factored form by starting with x to the third open parentheses, and now we divide each
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of these two terms by x to the third, leaving us with x to the first power, or just x, minus eight. This is the factored form
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of this polynomial. Now let's take a look at this polynomial. We have three terms here. So as we look for the GCF
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of these three terms, we again begin with the numerical coefficients.
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Let's make a list in this case of the factors of each of these numerical coefficients.
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The factors of twelve include one, two, three, four, six, and twelve. The factors of twenty
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include one, two, four, five, ten, and twenty. And the factors of ten
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include one, two, five, and ten. Let's find the factors that we have in common
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for these three numerical coefficients. Well, of course they all have a factor of one.
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I notice that they also have a factor of two in common. There are no other factors in this list
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that are common between all three numerical coefficients. So our numerical GCF is two.
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And now looking for the variable factors GCF, we have x squared, x to the first,
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and x to the third. So the GCF of this variable factor is x to the first power. We have a y to
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the third, y squared, and y to the fourth. So the GCF of this variable factor is y squared.
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So the GCF of the three terms of this polynomial is two x y squared. Now let's write this
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polynomial in factored form. We begin with the GCF two x y squared followed by an open parenthesis.
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And now we divide each of these terms by our GCF two x y squared. Twelve divided by two is six.
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x squared divided by x is x. And y to the third divided by y squared is y.
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Moving to our next term, negative twenty divided by two is negative ten. x divided by
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x is one. y squared divided by y squared is one. So we're just left with minus ten
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when we factor out the two x y squared from minus twenty x y squared. And now for the last term,
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we divide ten x cubed y to the fourth by two x y squared. Ten divided by two is five.
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x cubed divided by x is x squared. And y to the fourth divided by y squared is y squared.
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This is factored form of this polynomial. And in our next example, we have y
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times x minus four plus three times x minus four. Let's first identify the terms
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in this polynomial. We have the term y times x minus four and the term three times x minus four.
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I notice here that what they have in common is the factor x minus four.
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Each of these terms has a factor of x minus four in common, and that is the GCF of this polynomial. So let's factor
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out that GCF, x minus four. We write the GCF followed by an open parenthesis, and now we'll divide each term by that GCF,
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x minus four. If I divide y times x minus four by x minus four, that leaves me with y.
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And in the second term, if I divide three times x minus four by x minus four, that leaves me with
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three or plus three. So factored form of this polynomial is x minus four times y plus three.
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Next, we're going to talk about a type of factoring called factoring by grouping.
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Let's look at these two examples where we're asked to factor each polynomial.
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Sometimes we can factor a polynomial by grouping the terms of the given polynomial
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and looking for common factors in those groupings. This is known as factored by grouping.
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We often look for polynomials with four terms as possible candidates for this method. Let's consider this polynomial
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first with four terms, a b plus six a plus two b plus twelve. We'll begin by grouping this polynomial into groups
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of two. So I will group a b plus six a, as one group and two b plus twelve as the second group. Now let's look for
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the GCF of our first group, a b and six a. The GCF of the numerical coefficients is one. The GCF of the variable factors
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is a. So the GCF of this grouping is a. If I factor an a out of this first group, that will leave me with b plus six.
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Now I have a plus sign which we will include here, and we'll look for now the GCF of our second group, two b plus twelve.
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The GCF of the numerical coefficients is two, and there is no common factor of the variable factors.
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So the GCF of these two terms is two. We will now factor a two out of this grouping.
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And when I factor a two out of two b plus twelve, that leaves us with b plus six. Now this
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intermediate result looks familiar, similar to the example that we did in the last problem.
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Notice now we have this written as two terms, a times b plus six and two times b plus six.
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Each of these two terms has the common factor b plus six. So we now can factor the common factor b
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plus six out of this expression. So I will factor a b plus six out, and then open parentheses, and
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now we divide each term by that common factor to find out what remains. a times b plus six
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divided by b plus six is a, and two times b plus six divided by b plus six is plus two.
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This is the factored form of this original polynomial. It's important when we're doing grouping to look for ways
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that we can get a common factor from the two groupings. Sometimes it's necessary to rearrange terms
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or to factor out a different sign positive or negative for our GCF.
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Let's take a look at this next example, x cubed minus x squared minus five x plus five.
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We'll begin by grouping this into groups of two. But notice, that I have a minus sign
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in the front of my third term. So let's rewrite this first as x cubed minus x squared plus
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a negative x plus five. And now I'm going to group this polynomial into groups of two. The first
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grouping will be x cubed minus x squared, and my second grouping will be negative five x plus five.
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The plus sign still remains between these two groupings. Now we look for the GCF of x cubed and x squared,
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which is x squared. And if I factor an x squared out of x cubed minus x squared, that leaves me with x minus one.
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Now looking at this next grouping, I have negative five x plus five. The numerical coefficient that
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we have in common here is five, and there is no variable factor that we have in common. So
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five would be a potential candidate for our GCF. But notice what would happen if I factored out a
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positive five. This would leave me with a negative x plus one, and what I'm looking for is a positive
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x and a negative one. So instead of factoring out a positive five, let's factor out instead
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a negative five. Dividing each term here now by negative five leaves me with x minus one. And I see then that
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I have two terms with the common factor of x minus one. So we can again factor out the GCF of x minus one.
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x minus one is our common factor. And when I divide each term by x minus one,
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that will leave me with x squared minus five. This is factored form of the polynomial.