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Transcript of Lesson 3.09
Fun with Factoring! 1. What is a GCF and how do I find it? Don't sweat it! Factoring is easy once you get the hang of it. A GCF is the greatest common factor, or the greatest factor that given numbers have in common A GCF can be found in a few different ways Option 1 1.List the prime factors of each number.
2.Multiply those factors both numbers have in common.
3. If there are no common prime factors, the GCF is 1. Option 2 1. Make a list of the factors for each number and the greatest number they all have in common is the GCF.
2. If there are no common prime factors, the GCF is 1. Numbers: 7 and 14
Factors of 7= 1*7
Factors of 14=2*7
GCF=7 Option 3 1. Make a factor tree
2. If there are no common prime factors, the GCF is 1. For more info try this Khan Academy video Now that we know how to find a GCF, let's put it into practice 2. Using a GCF to factor a polynomial Let's try factoring 6x^2y+14xy^2-42xy-2x^2y^2 1. Find the GCF
2x^2y^2=2*x*x*y*y GCF=2xy 2. We divide each term by the GCF to get the final factored form (6x^2y)/2xy=3x
(2x^2y^2)=xy Final solution after factoring out the 2xy= 2xy(3x+7y-21-xy) Still confused? Watch for more info This expression is called a difference of two squares.
(Notice the subtraction sign between the terms.)
a^2-b^2 Factoring difference of squares The factors of this are (a+b)(a-b) How you ask? Notice, the square root of each perfect square (a^2 and b^2) is a and b. These terms are placed inside two sets of parentheses where one is addition and one is subtraction. Multiplying the two factors (a + b)(a - b) results in the following:
(a + b)(a - b)
a^2 + ab - ab - b^2
a^2 + 0ab - b^2
a^2 - b^2 Example 1:
Factor: x^2 - 9
Both x^2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares.
What times itself will give x^2 ? The answer is x.
What times itself will give 9 ? The answer is 3.
The factors are (x + 3) and (x - 3).
Answer: (x + 3) (x - 3) Example 2:
Factor:4x^2 - 16
First find the GCF. After making a list of factors, you will see that it is 4.
So, so far we have 4( )
Next, we have to divide both of the numbers by 4.
So, we now have 4(x^2-4)
Since the binomial inside the parentheses only has two terms that are separated by a subtraction sign, that it is a difference of squares.
What times itself will give us x^2? The answer is x.
What times itself will give us 4? The answer is 2.
The factors are: (x+2)(x-2)
Final Answer: 4(x+2)(x-2) Need some help? Check out this quick video! Perfect Square Factoring Formulas
a² - 2ab + b² = (a - b)²
a² + 2ab + b² = (a + b)²
So basically, "if" you are sure it is a perfect square formula, then you just have to locate "a" and "b". Here is an example:
25x² - 20x + 4
1) The first and last terms are perfect squares.
25x² - 20x + 4 = (5x)² - 20x + (2)²
2) The middle term is twice the product of the value 5x and the value 2. In other words, 2(5x)(2) = 20x.
(5x)² - 2(5x)(2) + (2)² = (5x - 2)²
Therefore, we can use the formula a² - 2ab + b² = (a - b)². Thus,
25x² - 20x + 4 = (5x - 2)² Factoring perfect square trinomials If you're in a pinch then this video might work Factoring trinomials Example 1: Factor the trinomial: y^2-5y+6
Note that this trinomial does not have a GCF.
So we go right into factoring the trinomial of the form .
Step 1: Set up a product of two ( ) where each will hold two terms.
It will look like this: ( )( )
Step 2: Find the factors that go in the first positions.
Since we have y squared as our first term, we will need the following:
(y )(y )
Step 3: Find the factors that go in the last positions.
We need two numbers whose product is 6 and sum is -5. That would have to be -2 and -3.
Putting that into our factors we get:
Step 4: To check you can multiply it back out and it is correct if you get the original trinomial as the answer. Factoring by grouping 3y^2 + 14y + 8
3y^2 + 14y + 8 = 3y^2 + 12y + 2y + 8
(3y^2 + 12y) + (2y + 8) = 3y(y + 4) + 2(y + 4)
So, 3y^2 + 14y + 8 =(y + 4)(3y + 2) To factor by grouping we must make it into a 4 term expression. Notice that 14y can be split into 12y+2y Find the GCF for each pair of binomials Group together the binomials in common (y+4) as one of the factors, and the two GCFs (3y+2) as the other factors. What if it has more than 3 terms? Don't fret, here's how to factor a 4 term polynomial. Example 6: Factor by grouping: x^3+7x^2+2x+14
Note how there is not a GCF for ALL the terms. So let’s go ahead and factor this by grouping.
Step 1: Group the first two terms together and then the last two terms together.
Step 2: Factor out a GCF from each separate binomial.
Step 3: Factor out the common binomial.
*Divide (x + 7) out of both parts
If we multiply our answer out, we do get the original polynomial. Check out this video for some more examples! Sum and difference of cubes A polynomial in the form a^3 + b^3 is called a sum of cubes. A polynomial in the form a^3 – b^3 is called a difference of cubes.
A sum of cubes:
A difference of cubes:
(a-b)(a^2+ab+b^2) Factor: x^3 + 125.
Final solution:(x+5)(x^2-5x+25) Factor: m^3-8
Final solution: (m-2)(m^2+2m+4) That's all for today's lesson. Happy calculations!