Factoring & Special Cases--- Week 13 11/4 Factoring Quadratics Factoring & Special Cases--- Week 13 11/4

Binomial Factors Every Quadratic can be simplified to the product of a constant and/or one or two binomial factors. Binomial means two terms, like (x – 3) or (2x +7) Binomial Factors of a quadratic will be linear if there are two. [mean x to the power of 1] like (x + 2) But could be in x2 if there is one and a constant: 5 (x2 +1) Factors are multiplied together (by F.O.I.L) to make a quadratic in standard form.

How to Factor a Quadratic For quadratic in standard form: a x2 + bx +c See if you can factor out a GCF (greatest common factor) Ex. In 4x2 - 8x - 12 factor out 4 -- 4(x2 - 2x -3) Next, set up 2 binomial factors in parentheses Ex. for (x2 -2x -3) make (x ) (x ) Determine the signs you will need You will use FACTORS of C and SUM to B, so two numbers you can multiply together to make the constant (c) and add up to the coefficient of x in the middle term (b). If c is positive, the values will be BOTH positive or BOTH negative. If c is negative, one will be positive, one negative. Ex. above: (x + ) (x - ) Find values as stated above – a pair of factors of c that add up to b. Be aware of signs! If c is negative you will be SUBTRACTING on factor from the other.

AC Method – if x2 term has a coefficient you cannot factor out. For something like: 4x2 – 20x +25 Multiply A times C: so 4 times 25 = 100 Find a factor pair of AC that adds up to B (watch signs!) So, factor pairs of 100 are: 100, 1 50,2 25,4 20,5 10,10 Rewrite quadratic with b term broken into two parts from factor pair used above: -10, -10 adds up to -20, so 4x2 -10x -10x +25 Separate quadratic into two parts and factor out GCF of each: 4x2 -10x -10x +25 2x (2x – 5) - 5 (2x – 5) *note, should always have the same factor on both sides Factors are the one that’s on both sides, and whatever is left (in this case, the same factor again!) (2x – 5) and (2x – 5)

Special Case Quadratics Perfect Square Trinomials Trinomial means 3 terms (like a standard form quadratic) Some quadratics are the sum of the SAME binomial twice (like the example on the previous slide). These are called perfect square trinomials. They can be the product of a sum or a difference: a2 + 2ab + b2 = (a + b)2 a2 -2ab + b2 = (a – b)2 This is the general form for these special cases – it’s a bit confusing because the a and b here are not exactly the a and b we see in the standard form. In the example from the last slide, to fit this form, “a” would be 2, not 4, because its in the form a2 + 2ab + b2 and so “b” would be 5 (square root of 25) Identifying these can save time! If we had seen that 4 and 25 were perfect squares, and that “b” was twice the product of these squares (2 (2x5)) we could have skipped the whole AC deal, and gone straight to (2x -5)2

Special Case Quadratics Difference of Squares This means “standard form” will be only a BINOMIAL! It will be in the form a2 – b2 a2 – b2 = (a + b) (a – b) Example: 9x2 – 81 is a difference of squares, since 9 and 81 are both perfect squares. The factored form would just be (3x – 9) (3x + 9) This works because with a “difference” , the “c” value is negative, so one term will be plus and one minus – meaning if they are the same, the whole middle “b” term will cancel! So, to be a difference of squares, a and c must both be perfect squares, there must be no “b” (no x to the first) term, and they MUST be subtracted!

Work Together p. 221 Lesson Check 1-13