5-4 Factoring Quadratic Expressions Perfect square trinomials

What is a perfect square trinomial? A perfect square trinomial is the product you get when you square a binomial (you multiply the binomial by itself) To square a binomial Example: (x+5) (x+5) square the first terms _____ x _____= _____ square the last terms _____ x _____= _____ the middle term is two times the product of the terms 2( _____ _____) = ______ The result is x2 + 10x + 25

Factoring a perfect square trinomial when all terms are positive 9x2 + 48x + 64 Take the square root of the first and third terms First term (3x) third term (8) Put the terms in the parenthesis squared and separate the terms with a plus sign. (3x + 8)2

4x2 + 24x + 36 Take the square root of the first and third terms and separate them by an addition sign (2x + 6)2 Check your answer: Square the first term (2x)2 = 4x2 Square the last term (6)2 = 36 Multiply the two terms together and double them (2(2x)(6)) =24x Try this one: 4x2 + 28x + 49

Factoring a perfect square trinomial when the middle term is negative 4n2 – 20n + 25 The only difference with factoring these trinomials is that the sign between the two square roots is negative Take the square root of the first and third terms First term (2n) third term (5) Separate the terms with a negative sign (2n - 5)2

4n2 – 16n + 16 Take the square root of the first and third terms and separate the numbers with a subtraction sign (2n - 4)2 Check your answer: Square the first term (2n)2 = 4n2 Square the last term (4)2 = 16 Multiply the two terms together and double them (2(2n)(-4)) =-16n Try this one: 9x2 - 42x + 49

Factoring the difference of two squares The difference of two squares is written as: a2 – b2 When they are factored they become (a+b)(a-b) When the terms of the binomials are FOIL’d the middle terms cancel each other out because one is positive and one is negative (a+b)(a-b) = a2 + ab – ab - b2 = a2 – b2 Take the square root of the first term and the square root of the second term and place them into two sets of parentheses – one set separated with a plus sign and one set separated with a minus sign

Example: c2 – 64 = (c+8)(c-8) Square root of the first term is c Square root of the last term is 8 Put the terms into two parentheses and separate one with a plus sign and one with a minus sign Remember: the middle terms will cancel out when the binomials of the difference of two squares are multiplied together.

Let’s try this one: 4x2 – 16 Take the square root of the first term: 2x Take the square root of the last term: 4 Rewrite the terms in two parentheses (2x 4) (2x 4) Separate the terms by a plus sign in one of the parentheses and a minus sign in the other parentheses (2x + 4) (2x – 4) How about this one: 9x2 – 36

One last note: Sometimes you may have to factor out the GCF before you can factor the quadratic. You can try to find factors of the first term and then find factors of the last term to make two binomials that you can multiply together If you can factor out a GCF first – do so in order to make the factoring easier 3n2 – 24n – 27 3(n2 – 8n – 9) 3(n – 9)(n + 1) This technique works for any trinomial you are trying to factor

h/w: p. 264: 38, 39, 41, 42, 44, 45, 52, 55, 57, 58