Factoring Kevin Ton Sam Wong Tiffany Wong

Trinomials Trinomials are written and found in the form of ax 2 +bx+c. In this slide, we will explore a trinomial that contains an a-value of 1. Example: (1x 2 +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 (1x 2 +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: 2x 3 +4x 2 +3x+ 6 Try to separate it with two sets of parentheses: (2x 3 +4x 2 )+(3x+ 6) Now, find the GCFs of each set of parentheses and then factor it out: 2x 2 (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 2x 2 and the 3. Since they are both the same, we can just include one to add to the new set of parentheses: (2x 2 +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: (x 2 -9)(x-2), you can further break down (x 2 -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 ax 2 +bx+c. Factoring a trinomial that includes an a-value larger than 1 is not that much harder. Example: (2x 2 +9x-5) The first position of the factors will be a product of 2x 2. The last positions will be the product of 5. The only way to get a product of 2x 2 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 – ax 2 + bx + c Example: x 2 + 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) = x 2 + 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: (a 2 -b 2 )  (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: x 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: (x 2 +9) (x 2 -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 a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) Example: 27x First focus on the variable b and find the cube root – 27x = (3x) Once the cube root is found, simply plug into the equation. – (3x + 1)[(3x) 2 – (3x)(1) ] Do a little cleaning up, and Done :] – (3x + 1)(9x 2 – 3x + 1) Difference of Cube a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) Example: x 3 – 8 First focus on the variable b and find the cube root – x 3 – 8 = x 3 – 2 3 Once the cube root is found, simply plug into the equation. – (x – 2)(x 2 + 2x ) Do a little cleaning up, and Done :] – (x – 2)(x 2 + 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!