MATH 31 LESSONS PreCalculus 1. Simplifying and Factoring Polynomials

A. Simplifying Polynomials When you simplify a polynomial, you are removing the brackets. e.g. (2x - 3) (4x + 1) = 8x x - 3 Also, you are reducing a polynomial to the smallest number of terms.

1. Adding and Subtracting Polynomials You can add or subtract monomials only with like terms. e.g. 5x + 7x = 12x 11y 2 - 7y 2 = 4y 2 6ab ab 3 = 17ab 3

If they are not like terms, then you cannot add them. e.g. 2x + 3y 5y 2 - 8y 3 12xy 2 + 8x 2 y

Ex. 1Simplify 2x - 11y + 7x + 3y + 5x Try this example on your own first. Then, check out the solution.

2x - 11y + 7x + 3y + 5x Identify the like terms

2x - 11y + 7x + 3y + 5x =2x + 7x + 5x - 11y + 7y Collect the like terms

2x - 11y + 7x + 3y + 5x =2x + 7x + 5x - 11y + 3y =14x - 8y

2. Multiplying Polynomials  Monomial  Monomial Consider 5a 2 b 3  10ab 4 =

5a 2 b 3  10ab 4 = (5  10) (a 2  a) (b 3  b 4 ) Multiply numbers and like variables separately

5a 2 b 3  10ab 4 = (5  10) (a 2  a) (b 3  b 4 ) =50 a 3 b 7

 Monomial  Polynomial Consider 5x (6x - 7) =

5x (6x - 7) = 5x (6x) - 5x (7) Multiply the monomial to each term of the polynomial

5x (6x - 7) = 5x (6x) - 5x (7) =30x x

 Binomial  Binomial Consider (2x - 3) (4x + 1) =

(2x - 3) (4x + 1) = 2x (4x) Use FOIL:First

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) Use FOIL:First Outside

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) - 3 (4x) Use FOIL:First Outside Inside

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) - 3 (4x) - 3 (1) Use FOIL:First Outside Inside Last

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) - 3 (4x) - 3 (1) =8x 2 + 2x - 12x - 3 =8x x - 3

 Polynomial  Polynomial Consider (x + 2y) (5x - 3y + 6) =

(x + 2y) (5x - 3y + 6) = x (5x) - x (3y) + x (6) Multiply the first term to the entire polynomial

(x + 2y) (5x - 3y + 6) = x (5x) - x (3y) + x (6) + 2y (5x) - 2y (3y) + 2y (6) Then, multiply the second term to the entire polynomial

(x + 2y) (5x - 3y + 6) = x (5x) - x (3y) + x (6) + 2y (5x) - 2y (3y) + 2y (6) = 5x 2 - 3xy + 6x + 10xy - 6y y = 5x 2 + 6x + 7xy - 6y y

Ex. 2Simplify 2 (3a + 4) (5a - 6) - (2a - 7) 2 Try this example on your own first. Then, check out the solution.

2 (3a + 4) (5a - 6) - (2a - 7) 2 =2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7) If it is a perfect square, then you should write both binomials. Then, you will remember to FOIL. Notice: (2a - 7) 2  (2a) 2 - (7) 2

2 (3a + 4) (5a - 6) - (2a - 7) 2 =2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7) =2 (15a a + 20a - 24) - (4a a - 14a + 49) Be certain to show the brackets around the entire product

2 (3a + 4) (5a - 6) - (2a - 7) 2 =2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7) =2 (15a a + 20a - 24) - (4a a - 14a + 49) =2 (15a 2 + 2a - 24) - (4a a + 49)

2 (3a + 4) (5a - 6) - (2a - 7) 2 =2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7) =2 (15a a + 20a - 24) - (4a a - 14a + 49) =2 (15a 2 + 2a - 24) - (4a a + 49) = 30a 2 + 4a a a - 49 Distribute the negative to all terms

2 (3a + 4) (5a - 6) - (2a - 7) 2 =2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7) =2 (15a a + 20a - 24) - (4a a - 14a + 49) =2 (15a 2 + 2a - 24) - (4a a + 49) = 30a 2 + 4a a a - 49 =26a a - 97 Add like terms

B. Factoring Polynomials When you factor a polynomial, you are adding brackets. e.g. 8x x - 3 = (2x - 3) (4x + 1) You are making a polynomial into a product.

1. Greatest Common Factor (GCF) The GCF is:  the largest number that divides evenly into the coefficients  the smallest power of each variable Taking out the GCF is usually the first step of factoring.

e.g. Factor 12 x 3 y x 8 y 2

12 x 3 y x 8 y 2 =6 x 3 y 2 ( The largest number that divides into 12 and 18 evenly The smallest power of each variable

12 x 3 y x 8 y 2 =6 x 3 y 2 ( 2 x 3-3 y x 8-3 y 2-2 ) When you factor (divide), you subtract the exponents

12 x 3 y x 8 y 2 =6 x 3 y 2 ( 2 x 3-3 y x 8-3 y 2-2 ) = 6 x 3 y 2 ( 2 x 0 y 2 + 3x 5 y 0 ) =6 x 3 y 2 ( 2 y 2 + 3x 5 )

2. Difference of Squares Formula: A 2 - B 2 = (A + B) (A - B) Note: There is no formula for A 2 + B 2.

e.g. Factor81 m y 6 z 4

81 m y 6 z 4 =(9 m 4 ) 2 - (4 y 3 z 2 ) 2 Put into the form A 2 - B 2.

81 m y 6 z 4 =(9 m 4 ) 2 - (4 y 3 z 2 ) 2 =(9 m y 3 z 2 ) (9 m y 3 z 2 ) A 2 + B 2 = (A + B) (A - B) where A = 9 m 4 and B = 4 y 6 x 2

3. Sum / Difference of Cubes Formulas: A 3 - B 3 = (A - B) (A 2 + 2AB + B 2 ) A 3 + B 3 = (A + B) (A 2 - 2AB + B 2 )

e.g. 1 Factorx y 3

x y 3 =(x) 3 - (4 y) 3 Put into the form A 3 - B 3

x y 3 =(x) 3 - (4 y) 3 =(x - 4y) [ x 2 + (x) (4y) + (4y) 2 ] A 3 - B 3 = (A - B) (A 2 + AB + B 2 ) where A = x and B = 4y

x y 3 =(x) 3 - (4 y) 3 =(x - 4y) [ x 2 + (x) (4y) + (4y) 2 ] =(x - 4y) (x 2 + 4xy + 16y 2 )

e.g. 2 Factor8x y 6

8x y 6 =(2x) 3 + (3 y 2 ) 3 Put into the form A 3 + B 3

8x y 6 =(2x) 3 + (3 y 2 ) 3 = (2x + 3y 2 ) [ (2x) 2  (2x) (3y 2 ) + (3y 2 ) 2 ] A 3 + B 3 = (A + B) (A 2 - AB + B 2 ) where A = 2x and B = 3y 2

8x y 6 =(2x) 3 + (3 y 2 ) 3 = (2x + 3y 2 ) [ (2x) 2  (2x) (3y 2 ) + (3y 2 ) 2 ] = (2x + 3y 2 ) (4x 2  6xy 2 + 9y 4 )

4. Grouping When there are 4 terms, try grouping:  Group pairs of terms (you may need to rearrange)  Factor each pair  Factor out the common polynomial

e.g. Factor ac  bd + bc  ad

ac  bd + bc  ad No common factors for each pair. Thus, we need to rearrange.

ac  bd + bc  ad =ac  ad + bc  bd

ac  bd + bc  ad =ac  ad + bc  bd =a (c  d) + b (c  d) They must have a common factor.

ac  bd + bc  ad =ac  ad + bc  bd =a (c  d) + b (c  d) =(a + b) (c  d)

5. Factoring Trinomials Trinomials are polynomials with 3 terms. They have the form Ax 2 + Bx + C = 0 We will deal with two cases: Case 1: A = 1 (By inspection) Case 2: A ≠ 1 (Decomposition)

Case 1: A = 1 (By inspection) To factor x 2 + Bx + C,  Find 2 numbers that add to B and multiply to C  Simply substitute the numbers into the two binomial factors

e.g. Factor x 2 + 2x - 15

x 2 + 2x - 15 Find two numbers that...add to 2

x 2 + 2x - 15 Find two numbers that...add to 2 and multiply to -15

x 2 + 2x numbers: Sum = 2 Product = -15 5, -3

x 2 + 2x numbers: Sum = 2 Product = -15 =(x + 5) (x - 3) Simply sub the numbers in 5, -3

Case 2: A ≠ 1 (Decomposition) To factor Ax 2 + Bx + C,  Find 2 numbers that add to B and multiply to AC  Replace B with these two numbers  Factor by grouping

e.g. Factor 3x x + 10

3x x + 10 Find 2 numbers: Sum = -17

3x x + 10 Find 2 numbers: Sum = -17 Product = 30

3x x + 10 Find 2 numbers: Sum = -17 Product = , -2

3x x + 10 =3x x - 2x + 10 Replace B with the two numbers, -2 and -15

3x x + 10 =3x x - 2x + 10 =3x (x - 5) - 2 (x - 5) Factor by grouping

3x x + 10 =3x x - 2x + 10 =3x (x - 5) - 2 (x - 5) =(x - 5) (3x - 2)

Summary (Factoring methods)  GCF first  Look at the # of terms: 2 terms : - Difference of squares - Sum / difference of cubes 3 terms: - Inspection (if A = 1) - Decomposition (if A ≠ 1) 4 terms: - Grouping

Ex. 3 Factor 80 xy xz 6 completely. Try this example on your own first. Then, check out the solution.

80 xy xz 6 =10x (8y 3 + z 6 ) Factor GCF first.

80 xy xz 6 =10x (8y 3 + z 6 ) Don’t stop here. Do you see what else can be factored?

80 xy xz 6 =10x (8y 3 + z 6 ) = 10x [ (2y) 3 + (z 2 ) 3 ]Sum of cubes

80 xy xz 6 =10x (8y 3 + z 6 ) = 10x [ (2y) 3 + (z 2 ) 3 ] =10x (2y + z 2 ) [ (2y) 2 - (2y) (z 2 ) + (x 2 ) 2 ]

80 xy xz 6 =10x (8y 3 + z 6 ) = 10x [ (2y) 3 + (z 2 ) 3 ] =10x (2y + z 2 ) [ (2y) 2 - (2y) (z 2 ) + (x 2 ) 2 ] =10x (2y + z 2 ) (4y 2 - 2yz 2 + x 4 )

Ex. 4 Factor x 2 y x 2 - 9y completely. Try this example on your own first. Then, check out the solution.

x 2 y x 2 - 9y We will factor by grouping (4 terms). However, we must rearrange so that there will be common factors. Can you see how?

x 2 y x 2 - 9y =x 2 y - 9y + 6x This is one way to do so.

x 2 y x 2 - 9y =x 2 y - 9y + 6x =y (x 2 - 9) + 6 (x 2 - 9)

x 2 y x 2 - 9y =x 2 y - 9y + 6x =y (x 2 - 9) + 6 (x 2 - 9) =(x 2 - 9) (y + 6) Don’t stop here. Can you see what else can be factored?

x 2 y x 2 - 9y =x 2 y - 9y + 6x =y (x 2 - 9) + 6 (x 2 - 9) =(x 2 - 9) (y + 6) =(x + 3) (x - 3) (y + 6)Difference of squares

Ex. 5 Factor 3a 4 - 7a completely. Try this example on your own first. Then, check out the solution.

Notice that 3a 4 - 7a is a trinomial. To make it easier to factor, let’s do a substitution. i.e. Let x = a 2 Then, 3 (a 2 ) (a 2 ) - 20 = 3x 2 - 7x - 20

3x 2 - 7x - 20 Find 2 numbers: Sum = -7 Product = , 5

3x 2 - 7x - 20 Find 2 numbers: Sum = -7 Product = -60 =3x x + 5x , 5

3x 2 - 7x - 20 Find 2 numbers: Sum = -7 Product = -60 =3x x + 5x - 20 =3x (x - 4) + 5 (x - 4) -12, 5

3x 2 - 7x - 20 Find 2 numbers: Sum = -7 Product = -60 =3x x + 5x - 20 =3x (x - 4) + 5 (x - 4) =(x - 4) (3x + 5) -12, 5

=(x - 4) (3x + 5) Finally, we have to back-substitute x = a 2 :

=(x - 4) (3x + 5) Finally, we have to back-substitute x = a 2 : =(a 2 - 4) (3a 2 + 5) Don’t stop here. Do you see what else can be factored?

=(x - 4) (3x + 5) Finally, we have to back-substitute x = a 2 : =(a 2 - 4) (3a 2 + 5) =(a + 2) (a - 2) (3a 2 + 5)