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Factoring. Kevin Ton Sam Wong Tiffany Wong. Trinomials. Trinomials are written and found in the form of a x 2 + b x+ c . In this slide, we will explore a trinomial that contains an a-value of 1. Example: ( 1 x 2 + 4 x+ 4 )
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Factoring Kevin Ton Sam Wong Tiffany Wong
Trinomials • Trinomials are written and found in the form of ax2+bx+c. • In this slide, we will explore a trinomial that contains an a-value of 1. • Example: (1x2+4x+4) • Using the reverse FOIL method, factoring this trinomial will result in a product of binomials or also known as factors. So it would look something like this: (x +or- __) (x +or- __). • The c-value would have to be a product of factors in the last position in each factor/set of parentheses. • So factoring the example trinomial. We must find sets of multiples that will equal 4 (the c-value). 1x4; 2x2 • Before we decide which set to use, we must also pay close attention to the b-value in the trinomial, 4. This is a key point to observe as it helps us decide which set of multiples will work. Looking at the set of multiples, we can easily figure out that 2 and 2 will equal 4, both from adding and multiplying them to each other. • Once we decide on the set of multiples, we can fill them into the last positions of the factor parentheses. (x +or- 2) (x +or- 2). • Finally, we need to figure out the signs of the factors within the parentheses (if it’s + or -). Looking at the trinomial, since there are two + signs, it is indicated that both factors will include + signs. As a result, the factors of (1x2+4x+4) are: (x+2)(x+2).
Trinomials: Tips • 1) Always look at the first variable, it will help when beginning a problem • Example 2x² + 9x − 5 • You know from the first variable 2x², that (2x+/- __) (x +/- __ ) • Answer (2x -1) (x+5) = 2x² + 9x − 5 • 2) Always look out for Greatest Common Factors for they will make the problem a lot easier! • Example 9y² + 7y – 2 • You know that from the first variable 9y² that you have (9y+/- __) (y +/- __ ). • Now multiply the first and last number together, while ignoring the signs 9 x 2 = 18. • With the GCF list all the factors that go in and use these to help with the parenthesis. • 18 has factors of • 1 x 18 • 2 x 9 • 3 x 6 • Now focus on the 7y and you know 9 – 2 = 7 and thus! • (9y + 9) (9y -2) Now reduce and you get (9y -2)(y + 1) .
Grouping • Factoring by grouping is fairly simple as you just need to focus on GCFs. • Example: 2x3+4x2+3x+ 6 • Try to separate it with two sets of parentheses: (2x3+4x2)+(3x+ 6) • Now, find the GCFs of each set of parentheses and then factor it out: 2x2(x+2)+3(x+2) • Here comes the easy part. Put the outside factored numbers into a new set of parentheses. Then observe the two factors of (x+2) that resulted from factoring out the 2x2 and the 3. Since they are both the same, we can just include one to add to the new set of parentheses: (2x2+3)(x+2), and ta-dah! You factored by grouping.
Grouping: Tips • 1) Always try to find the GCF • 2) Look for the greatest exponent in the trinomial and try to break it down. • If you end up with binomials like: (x2-9)(x-2), you can further break down (x2-9) using difference of squares: (x+3)(x-3)(x-2)
A-Values Larger than One • As explained in the Trinomials slide, trinomials are found in the form of ax2+bx+c. • Factoring a trinomial that includes an a-value larger than 1 is not that much harder. • Example: (2x2+9x-5) • The first position of the factors will be a product of 2x2. The last positions will be the product of 5. • The only way to get a product of 2x2 is to multiply 2x by x. And we also know that the only way to get a product of 5 is multiplying 1 by 5 or 5 by 1. • Now the only tricky part is to figure out which number, 1 or 5, goes with the 2x and which one will go with the x. • This is where you need to experiment a bit. Try out each combination while paying attention to the signs too. Make sure you get a c-value of-5 and a b-value of +9. • If you factor it correctly, you will get the binomials: (2x-1)(x+5).
A-Values Larger than One: Tips • Practice makes perfect! If the two numbers for the c-values do not work during the first try, try switching their positions. This is often the case when trying to match the b-value.
Of Quadratic Form • Trinomials in Quadratic Form will always be in • ax2 + bx + c • Example: x2 + 5x + 6 • In order to factor this Quadratic trinomial, we have to split the trinomial into two binomials • First look at 6 and decide two factors that could go into it • (2) and (3) because 2x3 = 6 and 2+3 =5 • Now insert the X’s (x+2) (x+3) = x2 + 5x + 6
Of Quadratic Form: Tips • Remember that a quadratic can be broken up into! • Two Binomials • Or a Polynomial and a Monomial • Always check for factors so that multiplied together they equal C and added equals B
Special Cases • Difference of Squares: (a2-b2) (a+b)(a-b) • A special type of factoring is difference of squares. It is considered “special” because most people would not think it is possible to factor the problem since a whole value (the b-value) is missing. It is missing because in the process of multiplying the two factors, the b-value cancels out or becomes 0 because of the opposite + and – signs, so it is not necessary to write the 0 in the expression or equation. • Example: x4-81 • We can use the same steps in factoring a trinomial for this case, the b-value will just equal zero. For that to happen, the last position in each factor must be the same value. In this case, it is 9 because they are factors of 81. • Answer: (x2+9) (x2-9)
Special Cases: Tips • The last position in the factors must be the same number and the signs of the factors must be the opposite, a plus and a minus sign.
Cubes • Sum of Cube a3 + b3 = (a + b)(a2 – ab + b2) • Example: 27x3 + 1 • First focus on the variable b and find the cube root • 27x3 + 1 = (3x)3 + 13 • Once the cube root is found, simply plug into the equation. • (3x + 1)[(3x)2 – (3x)(1) + 12] • Do a little cleaning up, and Done :] • (3x + 1)(9x2 – 3x + 1) • Difference of Cube a3 – b3 = (a – b)(a2 + ab + b2) • Example: x3 – 8 • First focus on the variable b and find the cube root • x3 – 8 = x3 – 23 • Once the cube root is found, simply plug into the equation. • (x – 2)(x2 + 2x + 22) • Do a little cleaning up, and Done :] • (x – 2)(x2 + 2x + 4),
Cubes: Tips • 1) Always begin with the B Variable in either equations! • 2) Remember that the signs change when using the two different equations! • One is called Sum and the other is called Difference for a reason!