Factoring by Grouping 04:05 minutes

Options
Video Transcript

Transcript Factoring by Grouping

Adventure Mike and his girlfriend plan to build a treehouse. Using a snake as a measuring stick, Mike figured out the polynomial expression to represent the total area.
Oh jeez, his girlfriend just remembered – she wants a balcony, so she and Mike can watch the sunset. What can he do? She’s the romantic type. So, to save time, rather than measuring and calculating everything all over again, he can use grouping to factor polynomials.

Standard Form of a Quadratic Polynomial

Let’s take a look at the expression he wrote, 15x²+9x-6. Hmm. This looks familiar, doesn’t it? This expression is in the standard form of a quadratic polynomial, ax²+bx+c, but notice it's a trinomial and 'a' is equal to a number other than 1. To figure out the measurements for the sides of the treehouse, how can Mike factor this expression?

Multiplying Binomials

To show him how to factor by grouping, let’s use another problem as an example. To put this method into perspective, we'll start at the end result, with the factors. Working backwards, first, use the FOIL method to multiply the two binomials. Before combining like terms, we have 3x² + 6x – x – 2. To help you understand factoring by grouping, pay attention to the terms that are highlighted. After combining these like terms, the result is a trinomial in standard, quadratic form with 'a' equal to a number other than 1.

Finding the factored form

Let’s work on Mike’s problem. Okay, so how do you get from the standard form to the factored form when 'a' is not equal to 1? There's a little trick to doing this. We need to find the factors of 'ac' that sum to 'b'. a = 15 and c = -6. So since 'ac' = -90, here's a list of some of the factors of -90 – let's take a look. Hmm can you find the pair of factors that also sum to 9? That's right. -6 and 15 are factors of -90 and sum to 9.
Now, watch carefully while I do some mathemagic. Write two new terms using the factors of 'ac' that sum to 'b', and multiplied by 'x'.
Does this format look familiar? Remember the highlighted terms - from the example problem? Now to group, use parentheses to group the four terms into two binomials. This is a little tricky because you have to group the terms so that, when you factor out the GCF, the remaining binomial is the same for each. Also, be especially careful you don't make a sign error. Last step, combine the two GCFs, to create a new binomial. The polynomial is factored!

Adventure Mike has the measurements for the two sides of the treehouse. If his girlfriend wants a bigger treehouse, it won’t be a problem because he can just adjust the size of the sides.

Let’s fast forward, the treehouse is finally finished. To remember the special moment, Adventure Mike snaps some photos. Let's look Oh no! It seems like Mike has been alone in the jungle - just a little too long…

Factoring by Grouping Übung

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

  • Explain how to factor the given quadratic polynomial by grouping.

    Tipps

    To factor quadratic polynomials by grouping, first multiply $a$ with $c$. Then sum all pairs of factors of this product and find the pair which sums to $b$.

    Working backwards, we have:

    $\begin{array}{rcl} (3x-1)(x+2)&=&3x(x+2)-(x+2)\\ &=&3x^2+6x-x-2 \end{array}$

    Lösung

    In order to factor such a quadratic polynomial,

    $ax^2+bx+c$,

    with $a$, $b$, and $c$ as coefficients, we must find all pairs of factors of $a\times c$.

    Then we search for the pair of factors whose sum is $b$.

    Let's have a look at an example: $3x^2+5x-2$.

    Here we have $a=3$, $b=5$, and $c=-2$ and thus $a\times c=3\times -2=-6$.

    The pairs of factors of $-6$ and their corresponding sums are:

    $\begin{array}{rr|r} \text{factor}&\text{factor}&\text{sum}\\ \hline -1&6&5\\ 1&-6&-5\\ -2&3&1\\ 2&-3&-1 \end{array}$

    The pair of factors which sums to $5$ is $-1$ and $6$.

    The following grouping

    $3x^2+5x-2=3x^2+6x-x-2$

    then gives us

    $3x(x+2)-(x+2)=(3x-1)(x+2)$.

    which is the factorization we are looking for.

  • Complete the following table of factors and sums.

    Tipps

    The product of a pair of factors must be $-90$. If you know one factor, just divide $-90$ by that factor to get the other factor in the pair.

    An example of a pair of factors of $-90$ and their sum:

    Lösung

    For $15x^2+9x-6$, we first multiply $15\times -6=-90$.

    We then know that we are looking for all pairs of factors of $-90$, and one pair in particular which sum up to $9$.

    Let's calculate the products and sums of all pairs of factors of $-90$:

    • $-1\times 90=-90$ and $-1+90=89$
    • $1\times (-90)=-90$ and $1-90=-89$
    • $-2\times 45=-90$ and $-1+90=89$
    • $2\times (-45)=-90$ and $2-45=-43$
    • $-3\times 30=-90$ and $-3+30=27$
    • $3\times (-30)=-90$ and $3-30=-27$
    • $-5\times 18=-90$ and $-5+18=13$
    • $5\times (-18)=-90$ and $5-18=-13$
    • $-6\times 15=-90$ and $-6+15=9$
    • $6\times (-15)=-90$ and $6-15=-9$
    We can now see that the pair $-6$ and $15$ satisfies our requirements: $-6\times 15=-90$ as well as $-6+15=9$.

  • Factor the polynomial $15x^2+9x-6$.

    Tipps

    The product of $15$ and $-6$ is $-90$.

    You can write $-90$ as a product of two factors. Find the pair of factors that add up to $9$.

    In the example,

    $\begin{array}{rcl} (2x+3)(4x-2)&=&2x(4x-2)+3(4x-2)\\ &=&8x^2-4x+12x-6\\ &=&8x^2+8x-6 \end{array}$

    we have that $8\times -6=48$, as well as $-4\times 12=48$. We can see that $-4+12=8$.

    First multiply $a$ and $b$. Then we check all pairs of factors of $ab$ whose product is $ab$. If we add a pair of factors together and get $b$, then we've found the pair we are looking for.

    Lösung

    To factor the quadratic trinomial $15x^2+9x-6$ with $a=15$, $b=9$, and $c=-6$, we first look for the pairs of factors of $a\times c$ that sum to $b$.

    We have that $a\times c=-90$, so we investigate all pairs of factors of $-90$:

    ${\small\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c} \text{Factors of }-90&-1,90&1,-90&-2,45&2,-45&-3,30&3,-30&-5,18&5,-18&-6,15&6,-15 \\ \hline \text{Sum of factors }&89&-89&43&-43&27&-27&13&-13&9&-9 \end{array}}$

    The sum of the factors $-6$ and $15$ is $-6+15=9$. So this is the pair we want.

    Next we group the quadratic polynomial as follows:

    $15x^2+9x-6=15x^2-6x+15x-6$.

    Grouping the binomials together, we get $(15x^2-6x)+(15x-6)$.

    The greatest common factor of the left binomial is $3x$ and of the right binomial is $3$:

    $(15x^2-6x)+(15x-6)=3x(5x-2)+3(5x-2)$.

    Now we see that both summands have the binomial $5x-2$ as a common factor, and thus we can factor even further:

    $3x(5x-2)+3(5x-2)=(3x+3)(5x-2)$.

  • Consider the appropriate factorization.

    Tipps

    You can either factor by grouping or use the FOIL method for binomials.

    An example of the FOIL method:

    $\begin{array}{rcl} (5x+1)(x+5)&=&5x^2+5x(5)+1x+1(5)\\ &=&5x^2+26x+5 \end{array}$

    Remember: First, Outer, Inner, Last.

    If you'd like to factor $ax^2+bx+c$ by grouping then:

    • Find all pairs of factors of $a\times c$.
    • Check which pair sums to $b$.

    Lösung

    To match the quadratic polynomial in standard form with its factorization, you can either factor the quadratic polynomial or multiply the binomials.

    Let's go over how to use both methods with the polynomial $2x^2-11x+12$.

    Factoring the Polynomial:

    First we note that $a=2$, $b=-11$ and $c=12$.

    First multiply $a$ and $c$ to get $2\times 12=24$.

    Next look for the pair of factors of $24$ which sum to $-11$; the pair we are looking for is $-8$ and $-3$.

    Then group to get $2x^2-11x+12=(2x^2-8x)-(3x-12)=2x(x-4)-3(x-4)$.

    Then factor out the greatest common factors of the binomials to get $2x(x-4)-3(x-4)=(2x-3)(x-4)=(-2x+3)(-x+4)$.

    Lastly, multiply each binomial with $-1$.

    Multiplying the Binomials:

    You can also use FOIL multiplication with $(3x-4)(2x-2)$.

    We have that $(3x-4)(2x-2)=6x^2-6x-8x+8=6x^2-14x+8$.

  • Indicate the coefficients of the quadratic polynomials.

    Tipps

    Keep in mind that the $-$ sign belongs to the coefficient.

    For the polynomial $3x^2+2x+4$, we have that $a=3$, $b=2$, and $c=4$.

    Pay attention: Neiter $x^2$ nor $x$ belong to $a$ or $b$.

    Lösung

    The standard form of quadratic polynomial is given by

    $ax^2+bx+c$, where $a\neq 1$.

    Before we start to group quadratic polynomials we have to note the values of $a$, $b$, and $c$.

    For the given polynomials, we look for the factor of the quadratic term $x^2$, $a$, the linear term, $b$, and the constant term, $c$:

    • For $6x^2+x-2$, we have $a=6$, $b=1$, and $c=-2$.
    • For $9x^2-25x-6$, we have $a=9$, $b=-25$, and $c=-6$.
    • For $8x^2+4x-12$, we have $a=8$, $b=4$, and $c=-12$.
    • For $8x^2+14x+3$, we have $a=8$, $b=14$, and $c=3$.
  • Factor the given polynomial.

    Tipps

    $ax^2+bx+c$ is the standard form of a quadratic polynomial.

    For example, with $x^2-x+1$, we have that $a=1$, $b=-1$, and $c=1$.

    Look for the pair of factors of $a\times c$ that sum to $b$.

    Once you group a polynomial into two binomials, each binomial has a greatest common factor. So you can factor the greatest common factor from each binomial.

    Lösung

    To factorize the trinomial $6x^2+x-2$ we first look at the coefficients: Here we have $a=6$, $b=1$ and $c=-2$.

    We're looking for the pairs of factors of $6\times (-2)=-12$ which sum to $1$.

    $\begin{array}{rr|r} \text{factor}&\text{factor}&\text{sum}\\ \hline -1&12&11\\ 1&-12&-11\\ -2&6&4\\ 2&-6&-4\\ -3&4&1 \end{array}$

    Now we can stop: We've found the factors $-3$ and $4$.

    We can rewrite $6x^2+x-2$ as follows:

    $6x^2+x-2=(6x^2-3x)+(4x-2)$.

    Just look at the two resulting binomials in parentheses. The greatest common factor of the first one is $3x$ and of the second one is $2$:

    $(6x^2-3x)+(4x-2)=3x(2x-1)+2(2x-1)$.

    Now we see that both summands have the binomial $2x-1$ as a factor in common. So we factor to get, $3x(2x-1)+2(2x-1)=(3x+2)(2x-1)$.

    The side lengths Adventure Mike was looking for are then $3x+2$ and $2x-1$.