Section 5.5 Factoring Polynomials Integrated Math Section 5.5 Factoring Polynomials

Prime number- a natural number that has only two factors (one and itself) Greatest common factor – is a monomial that includes every factor that is a factor of all the terms of the polynomial

http://www.youtube.com/watch?v=cRz4hW9SPPc

Finding the GCF Break up each term into prime factors Box in any factor that is common to all the terms Factor out the common factor(s)

Another way to find the GCF Write the prime factorization of each number Box in common factors-their product is the GCF 18 27 3∙3∙2 3∙3∙3 GCF = 3∙3=9

#1 Find the GCF of 28 and 40. #2 Find the GCF of 65, 150 and 200.

Find the GCF of 24𝑥 2 𝑦, 42𝑥𝑦 2 , 66𝑥𝑦 3 Break down to primes Box common factors Write the product

When you use distributive property, you are multiplying. 3𝑥 𝑥−5 =3 𝑥 2 −15𝑥

If a polynomial has more than one term, look for a GCF to factor out. Think of this as the inverse of distributive property! 𝐹𝑎𝑐𝑡𝑜𝑟 3 𝑥 2 −15𝑥 3𝑥𝑥−3∙5𝑥 3𝑥(𝑥−5)

You multiply with distributive property When you factor out a GCF, you divide!

Factor #1 10 𝑥 3 −20 𝑥 2 #2 25𝑥 3 − 20𝑥 2 +15𝑥 Factor out the opposite of the GCF #3 −27 𝑦 5 +45 𝑦 3 ( Find the GCF 1st!)

Plus and minus signs inside parentheses are not counted when determining terms! 6 𝑥 2 𝑥−2 +12𝑥(𝑥−2) What do these terms have in common? Are they factored down to primes?

Sometimes one of the common factors can be a binomial!

Factor these! #1 20𝑥 𝑥+10 +11(𝑥+10) #2 3𝑥 𝑥+8 −7 𝑥+8 #3 4𝑥 𝑥−2 −(𝑥−2)

When there is nothing to factor out of a four term polynomial, try factoring by grouping Factor first two terms, then the last two terms Do you have a binomial that can be factored out? Keep going!

Try factoring by grouping Try factoring by grouping! (four terms usually means this is a good path) 2𝑎+2𝑏+𝑤𝑎+𝑤𝑏

Watch out when the 3rd term is negative!!! Factoring out a negative changes the signs in the parentheses!

Caution Factor 2 𝑥 3 +6 𝑥 2 −3𝑥−9 2𝑥𝑥𝑥+2∙3𝑥𝑥−3𝑥−3∙3

#1 Factor by grouping: 𝑥𝑦−𝑦−6𝑥+6 #2 Factor by grouping: 4𝑎+4𝑏−𝑐𝑎−𝑐𝑏

Assignment #22A Pg. 340 #3-39 (x3)

Perfect Squares What is a perfect square number? What about a variable raised to a power?

Use FOIL to multiply: #1 (𝑥+3)(𝑥−3) #2 ( 𝑚 2 +5)( 𝑚 2 −5) #3 𝑥+𝑦 𝑥−𝑦 Do you see a pattern?

The difference of two perfect squares can be factored into two linear factors! #1 𝑥 2 −9 #2 16 𝑥 4 −1 #3 25 𝑥 2 − 𝑦 2 #4 𝑥 2𝑚 − 𝑦 2

The sum of two perfect squares does not factor into two binomials. Do not try to factor 𝑥 2 +9!!!!

There are perfect square trinomials that can be factored!! Look for a perfect square first term and perfect square last term

If so, proceed! Check the middle term using FOIL. Does it work out? If yes, the trinomial factors!!!

Factor 𝑥 2 −18𝑥+81 1st and last terms perfect squares? Middle term 2 times the product of the square root of 𝑥 2 and the square root of 81? Make two parentheses! What will determine the signs?

Try these!! #1 𝑥 2 +12𝑥+36 #2 4 𝑥 2 −24𝑥𝑦+ 36𝑦 2

Always look for a GCF to factor first!!! Check to see if any factors will factor again!!! Don’t stop until the polynomial is factored completely!

Factor these completely! #1 2𝑥 3 −50𝑥 #2 𝑥 2 2𝑥−1 −4(2𝑥−1) #3 2𝑏𝑛 2 − 4𝑏 2 𝑛+ 2𝑏 3

Assignment #22B Pg. 340 #41-47 odd, 51-81 (x3)