9.1 Adding and Subtracting Polynomials

Monomial Is an expression that is a number, variable, or a product of a number and one or more variables. Ex: 3, y, -4x2y, c/3

c/3 is a monomial, but c/x is not because there is a variable in the denominator.

Degree of a Monomial Is the sum of the exponents of its variables. For nonzero constant the degree is zero.

Ex 1: Find the degree of the monomial. The degree of a nonzero constant is __.

Ex 1: Find the degree of the monomial. The degree of a nonzero constant is _0_.

b) 3xy3 Degree ___ The exponenta are ___ and ___. Their sum is ___

b) 3xy3 Degree _4_ The exponents are _1_ and _3_. Their sum is _4_

c) 6c Degree: __ 6c = 6c1. The exponent is ___.

c) 6c Degree: _1_ 6c = 6c1. The exponent is _1_.

Polynomial Is a monomial or the sum or difference of two or more monomials. Ex: 4y + 2, 2x2 – 3x +1

In the Standard Form of a Polynomial The degrees of the monomials terms decrease from left to right.

Degree of a Polynomial Is the same as the degree of the monomial with the greatest exponent.

Binomial Is a polynomial of two terms. Ex: 3c + 4

Trinomial Is a polynomial of three terms. Ex: 2x4 + x2 - 2

Adding Horizontally Ex: (5x2 + 3x +4) + (3x2 + 5) 8x2 + 3x +9

Ex: (7a2b3 + ab) +(1 – 2a2b3) 5a2b3 + ab +1

Adding Polynomials in Columns

(2x4 – 5x2 + 4x + 5)+ (5x4 + 7x3 – 2x2 –2x)

2x4 + 0x3 – 5x2 + 4x + 5 5x4 + 7x3 – 2x2 – 2x + 0

2x4 + 0x3 – 5x2 + 4x + 5 5x4 + 7x3 – 2x2 – 2x + 0 7x4 + 7x3 – 7x2 + 2x + 5

Subtracting of Polynomials

Review : 2 polynomials are the additive inverses of each other if there sums equal zero.

4x7 – 7x - 5 Can be rewritten as 4x7 + (-7x) + (-5)

Key Questions Is x-3 the additive inverse of x3? No

Is x-3 the additive inverse of –x3? NO

Is x-3 the additive inverse of –x-3? Yes

Ex 1: Find the Additive Inverse 7x4 - 3x + 5 -7x4 + 3x – 5 Notice the signs changed

Change the sign of the terms in the 2nd parenthesis. Subtract (5x2 + 3x – 2) – (2x2 + 1) Change the sign of the terms in the 2nd parenthesis.

(remember you are subtracting a 1) (5x2 + 3x – 2) – (2x2 + 1) 5x2 + 3x – 2 – 2x2 – 1 Subtract like terms. (remember you are subtracting a 1)

Subtract (5x2 + 3x – 2) – (2x2 + 1) 5x2 + 3x – 2 – 2x2 – 1 Subtract like terms.

Subtract (5x2 + 3x – 2) – (2x2 + 1) 5x2 + 3x – 2 – 2x2 – 1

Subtract (2a2b2 + 3ab3 – 4b4) – (a2b2 – 5ab3 + 3b – 2b4) Change the sign of every term in the 2nd parenthesis.

Subtract (2a2b2 + 3ab3 – 4b4) – (a2b2 – 5ab3 + 3b – 2b4)

Subtract (2a2b2 + 3ab3 – 4b4) – (a2b2 – 5ab3 + 3b – 2b4) Subtract like terms. 2a2b2 + 3ab3 – 4b4 – a2b2 + 5ab3 – 3b + 2b4

a2b2 + 8ab3 – 2b4 – 3b

Use Columns to Subtract 8x3 + 6x2 – 3x + 5 minus 5x3 – 3x2 – 2x + 4

8x3 + 6x2 – 3x + 5 Change the sign of each term in the 2nd part.

8x3 + 6x2 – 3x + 5 -5x3 + 3x2 + 2x – 4 3x3 + 9x2 – x + 1

Subtract using columns 2a4b + 5a3b2 – 4a2b3 minus 4a4b +2a3b2 – 4ab

2a4b + 5a3b2 – 4a2b3 + 0ab -4a4b – 2a3b2 – 0a2b3 + 4ab -2a4b + 3a3b2 – 4a2b3 + 4ab

9 – 2 Multiplying and Factoring

Multiplying a Monomial and a Trinomial Ex 1: Simplify -2g3(3g3 + 6g – 5) Distribute -2g3 to each term inside the parenthesis.

( )(3g3) -2g2 ( ) –2g2 ( ) -2g2 (3g3) -2g2 (6g) –2g2(-5)

( )(3g3) -2g2 ( ) –2g2 ( ) -2g2 (3g3) -2g2 (6g) –2g2(-5) Remember when multiplying you add exponents.

( )(3g3) -2g2 ( ) –2g2 ( ) -2g2 (3g3) -2g2 (6g) –2g2(-5) Remember when multiplying you add exponents. -6g2 + – 12g2 + +

( )(3g3) -2g2 ( ) –2g2 ( ) -2g2 (3g3) -2g2 (6g) –2g2(-5) Remember when multiplying you add exponents. -6g2 + 3 – 12g2+1 + 10g2

Finding the Greatest Common Factor GCF – what monomial is a factor in each term.

Ex 2: Find the GCF of 2x4 + 10x2 - 6x List the prime factors of each term.

2x4 = 2• x • • x • 10x2 = 2• • x • 6x = 2• x

2x4 = 2• x • x • x • x 10x2 = 2• x • x • x 6x = 2 • 3 • x Which prime factors are in term?

2x4 = 2• x • x • x • x 10x2 = 2• x • x • x 6x = 2 • 3 • x Which prime factors are in term? A ‘2’ and an ‘x’.

The GCF is 2x.

Do QC #2a – c, Pg. 501.

Factoring Out a Monomial Find the GCF and then factor out the GCF from each term.

Factoring Out a Monomial Factor Ex: Factor 4x3 +12x2 – 16x First find the GCF.

List the prime factors. 4x3 = 12x2 = 16x =

List the prime factors. 4x3 = 2• •x• • • 12x2 = 2 • 2 • • x • x

List the prime factors. 4x3 = 2• 2 •x• x • x •

List the prime factors. 4x3 = 2• 2 •x• x • x • The GCF is 4x.

4x3 +12x2 – 16x Now Factor Out the GFC. 4x( ) + ( )(3x) + 4x(-4) 4x( x2 ) + ( 4x )(3x) + 4x(-4)

( ) (x2 + - ) (4x) (x2 + 3x - 4 )

Ex: 5x3 + 10 5 goes into each term 5(x3) + 5(2) 5(x3 + 2)

Ex: 6x3 + 12x2 6x2 goes into each term. 6x2 (x) + 6x2 (2) 6x2 (x + 2)

Ex: 12u3v2 + 16uv4 4uv2 goes into each term. 4uv2(3u2) + 4uv2(4v2) 4uv2(3u2 + 4v2)

6y2(3y2 – y + 2) Ex: 18y4 – 6y3 + 12y2 6y2 goes into each term. 6y2(3y2) - 6y2(y) + 6y2(2) 6y2(3y2 – y + 2)

Ex: 8x4y3 + 6x2y4 2x2y3(4x2) + 2x2y3(3y) 2x2y3(4x2 + 3y) 2x2y3 goes into each term. 2x2y3(4x2) + 2x2y3(3y) 2x2y3(4x2 + 3y)

There is no common factor. Ex: 5x3y4 + 7x2z3 + 3y2z There is no common factor.

9 – 3 Multiplying Binomials

Using the Distributive Property Ex 1: (4y + 5)(y + 3)

Using the Distributive Property Ex 1: (4y + 5)(y + 3) 4y(y + 3) + 5(y + 3)

Using the Distributive Property Ex 1: (4y + 5)(y + 3) 4y(y + 3) + 5(y + 3) 4y2 + 12y + 5y + 15

Using the Distributive Property Ex 1: (4y + 5)(y + 3) 4y(y + 3) + 5(y + 3) 4y2 + 12y + 5y + 15 Combine like terms

Using the Distributive Property Ex 1: (4y + 5)(y + 3) 4y(y + 3) + 5(y + 3) 4y2 + 12y + 5y + 15 Combine like terms 4y2 + 17y + 15

Use the Distributive Property Ex 2: (3a – b)(a + 7) 3a(a + 7) – b(a + 7) 3a2 + 21a – ab – 7b No like terms

Multiplying 2 Binomials When multiplying 2 Binomials, a useful tool to use is the FOIL Method.

First terms Outside terms Inside terms Last terms

Ex: (x + 5)(x + 4)

First terms (x + 5)(x + 4) x2

Outside terms (x + 5)(x + 4) 4x

Inside terms (x + 5)(x + 4) 5x

Last terms (x + 5)(x + 4) 20

x2 + 4x + 5x + 20 x2 + 9x + 20

Ex: (x + 2)(x + 3) x2 + 3x + 2x + 6 x2 + 5x + 6

Ex: (3x + 2)(x + 5) 3x2 + 15x + 2x + 10 3x2 + 17x + 10

Ex: (4ab + 3)(2ab2 +1) 8a2b3 + 4ab + 6ab2 + 3

Will there always be like terms after multiplying binomials?

Using the Vertical Method Ex 1: (7a + 5)(2a – 7)

9 – 4 Multiplying Special Cases

Multiplying a Sum and a Difference.

Product of (A + B) and (A - B)

Notice the middle terms eliminate each other! 5) Multiply (x – 3)(x + 3) Notice the middle terms eliminate each other! x2 First terms: Outer terms: Inner terms: Last terms: Combine like terms. x2 – 9 x -3 +3 +3x -3x x2 -3x -9 +3x -9 This is called the difference of squares.

Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2 You can only use this rule when the binomials are exactly the same except for the sign. (x – 3)(x + 3) a = x and b = 3 (x)2 – (3)2 x2 – 9

Ex: Multiply: (y – 2)(y + 2) Ex: Multiply: (5a + 6b)(5a – 6b) (5a)2 – (6b)2 25a2 – 36b2

Multiply (4m – 3n)(4m + 3n) 16m2 – 9n2 16m2 + 9n2 16m2 – 24mn - 9n2

The product of the sum and difference of two terms is the square of the first term minus the square of the second term.

(A + B)(A – B) A2 – B2

When multiplying the sum and difference of two expressions, why does the product always have at most two terms?

The other terms are additive inverses of each other and cancel each other out.

EX: (r + 2)(r – 2) r2 – 2r + 2r – 4 r2 – 4

Ex: (2x + 3)(2x – 3) 4x2 – 6x + 6x – 9 4x2 – 9

Ex: (ab + c)(ab – c) a2b2 – c2

Ex: (-3x + 4y)(-3x – 4y) 9x2 – 16y2

Squaring Binomials The square of a binomial is the square if the first term, plus or minus twice the product of the two terms, plus the square of the last term.

(A + B)2 = A2 + 2AB + B2 (A - B)2 = A2 - 2AB + B2

Common Mistake

Common Mistake

Ex:1) Multiply (x + 4)(x + 4) Notice you have two of the same answer? x2 First terms: Outer terms: Inner terms: Last terms: Combine like terms. x2 +8x + 16 x +4 +4x +4x x2 +4x +16 +4x +16 Now let’s do it with the shortcut!

1) Multiply: (x + 4)2 using (a + b)2 = a2 + 2ab + b2 That’s why the 2 is in the formula! 1) Multiply: (x + 4)2 using (a + b)2 = a2 + 2ab + b2 a is the first term, b is the second term (x + 4)2 a = x and b = 4 Plug into the formula a2 + 2ab + b2 (x)2 + 2(x)(4) + (4)2 Simplify. x2 + 8x+ 16 This is the same answer!

Ex: 2) Multiply: (3x + 2y)2 using (a + b)2 = a2 + 2ab + b2 a = 3x and b = 2y Plug into the formula a2 + 2ab + b2 (3x)2 + 2(3x)(2y) + (2y)2 Simplify 9x2 + 12xy +4y2

Multiply (2a + 3)2 4a2 – 9 4a2 + 9 4a2 + 36a + 9 4a2 + 12a + 9

Ex: (x + 5) 2 (x2 + 2(x)(5) + 25) x2 + 10x + 25

Ex Multiply: (x – 5)2 using (a – b)2 = a2 – 2ab + b2 Everything is the same except the signs! (x)2 – 2(x)(5) + (5)2 x2 – 10x + 25 Ex Multiply: (4x – y)2 (4x)2 – 2(4x)(y) + (y)2 16x2 – 8xy + y2

Ex: (y – 3)2 y2 - 2(y)(3) + 9 y2 – 6y + 9

Ex: (2a – 3b)2 4a2 - 2(2a)(3b) + 9b2 4a2 - 12ab + 9b2

Multiply (x – y)2 x2 + 2xy + y2 x2 – 2xy + y2 x2 + y2 x2 – y2

9 – 5 Factoring trinomials of the type x2 + bx +c

To factor a trinomial of in the form of x2 + bx + c Where c > 0

B.) their sum or difference is b. We must find 2 numbers that: A.) their product is c and B.) their sum or difference is b.

C. If the sign of c is (+), both numbers must have the same sign.

To figure out the 2 factors, we are to use the Try, Test, and Revise Method.

What two numbers have the sum of 5 and the product of 6? We look at the factors of 6 that add up to 5.

Factors of 6 1 and 6 2 and 3

Factors of 6 1 and 6 sum is 7 2 and 3 sum is 5 2,3

What two numbers have the sum of 8 and the product of 12? We look at the factors of 12 that add up to 8.

1 and 12 2 and 6 3 and 4

1 and 12 sum is 13 2 and 6 sum is 8 3 and 4 sum is 7 2,6

What two numbers have the sum of -8 and the product of 7? Since the sum is (-) and the product is (+). Both numbers are (-).

-1 and -7 Only set of factors.

-1 and -7 sum is -8

What factors of 12 add up to 8? x2 + 8x + 12 b = 8, c = 12 What factors of 12 add up to 8?

Factors of 12 1 and 12 -1 and -12 2 and 6 -2 and -6 3 and 4 -3 and -4

2 and 6 (x + 2)(x + 6)

What factors of 16 add up to -10? x2 – 10x + 16 b = -10, c = 16 What factors of 16 add up to -10?

Factors of 16 1 and 16 -1 and 16 2 and 8 -2 and -8 4 and 4 -4 and -4

-2 and -8 (x – 2)(x – 8)

What factors of 2 add up to -3? p2 – 3pq + 2q2 b = -3, c = 2 What factors of 2 add up to -3?

Factors of -2 1 and 2 -1 and -2 -1 and -2 (p – q)(p – 2q)

To factor a trinomial of in the form of x2 + bx + c Where c < 0

Watch your signs most students have the right factors, but the wrong signs.

What 2 numbers have a sum of -6 and a product of -7? -1 and 7 or 1 and -7 1 and -7

What 2 numbers have a sum of -2 and a product of -8? -1 and 8 1 and -8 -2 and 4 2 and -4 -2 and 4

What 2 numbers have a sum of 1 and a product of -2? 1 and -2 -1 and 2

What 2 numbers have a sum of 1 and a product of -2? 1 and -2 -1 and 2

Factor u2 – 3uv - 10v2 b = -3 and c = -10 What factors of -10 have the sum of -3.

Factors of -10 1 and -10 -1 and 10 2 and -5 -2 and 5

2 and -5 (u + 2v)(u – 5v)

Factor x2 + 3x - 4 b = 3 and c = -4 What factors of -4 have the sum of 3.

Factors of -4 1 and -4 -1 and 4 2 and -2

-1 and 4 (x – 1)(x + 4)

Factor y2 – 12yz – 28z2 b = -12 and c = -28 What factors of -28 have the sum of -12.

Factors of -28 1 and -28 -1 and 28 2 and -14 -2 and 14

2 and -14 (x + 2z)(x – 14z)

Review: (y + 2)(y + 4) Multiply using FOIL or using the. Box Method Review: (y + 2)(y + 4) Multiply using FOIL or using the Box Method. Box Method: y + 4 y y2 +4y + 2 +2y +8 Combine like terms. FOIL: y2 + 4y + 2y + 8 y2 + 6y + 8

What are the factors of y2? 1) Factor. y2 + 6y + 8 Put the first and last terms into the box as shown. y2 + 8 What are the factors of y2? y and y

1) Factor. y2 + 6y + 8 Place the factors outside the box as shown. What are the factors of + 8? +1 and +8, -1 and -8 +2 and +4, -2 and -4

1) Factor. y2 + 6y + 8 Which box has a sum of + 6y? + 1 y2 + 8 y + 2 y + y + 2y + 8y + 4y + 8 + 4 The second box works. Write the numbers on the outside of box for your solution. y

Here are some hints to help you choose your factors. 1) Factor. y2 + 6y + 8 (y + 2)(y + 4) Here are some hints to help you choose your factors. 1) When the last term is positive, the factors will have the same sign as the middle term. 2) When the last term is negative, the factors will have different signs.

What are the factors of x2? 2) Factor. x2 - 2x - 63 Put the first and last terms into the box as shown. x2 - 63 What are the factors of x2? x and x

2) Factor. x2 - 2x - 63 Place the factors outside the box as shown. What are the factors of - 63? Remember the signs will be different!

The second one has the wrong sign! 2) Factor. x2 - 2x - 63 Use trial and error to find the correct combination! x - 3 x2 - 63 x - 7 + 9 -7x +9x x x2 -3x + 21 +21x - 63 Do any of these combinations work? The second one has the wrong sign!

2) Factor. x2 - 2x - 63 Change the signs of the factors! + 7 +7x - 9 -9x Write your solution. (x + 7)(x - 9)

9 – 7 Factoring Special Cases

In 9 – 4 we used the square of binomials. (A + B)2 = A2 + 2AB + B2 (A - B)2 = A2 - 2AB + B2

EX: (x + 8)2 or (x - 8)2 A = x and B = 8 A2 + 2AB + B2 (x)2 + 2(x)(8) + (8)2 x2 + 16x + 64 A = x and B = 8 A2 - 2AB + B2 (x)2 - 2(x)(8) + (8)2 x2 - 16x + 64

Perfect-Square Trinomials A2 + 2AB + B2 =(A + B)(A + B) = (A + B)2 A2 - 2AB + B2 = (A - B) (A - B) = (A - B)2

The perfect-square trinomials are the reverse of the square binomials.

Ex 1: Where ‘A’ = 1 x2 + 10x + 25 A2 + 2AB + B2 A2 = x2 2AB = 10x B2 = 25 A = x and B = 5 (A + B)2 (x + 5)2

x2 + 10x + 25 = (x + 5)2

Always make sure the 1st and last terms are perfect squares. Ex 2: y2 – 22y + 121 Are y2 and 121 perfect squares?

Yes, both are perfect squares for y and 11 Since the middle term is negative, you subtract. (y – 11)2

Perfect squares? m2  m and 9n2  3n (m + 3n)2 Ex 3: m2 + 6mn + 9n2 Perfect squares? m2  m and 9n2  3n (m + 3n)2

64y2 + 48y + 9 Are 64y2 and 9 perfect squares? YES Ex 2: if A ≠ 1 64y2 + 48y + 9 Are 64y2 and 9 perfect squares? YES

The one thing you need to look at the middle term, is it equal to 2AB. (8y)2 + 48y + (3)2 (8y + 3)2 The one thing you need to look at the middle term, is it equal to 2AB. 2(8y)(3) = 48y

(4h)2 + 40h + (5)2 Does 2AB = 40h? 2(4h)(5) Yes, 40h Ex: 16h2 + 40h + 25 (4h)2 + 40h + (5)2 Does 2AB = 40h? 2(4h)(5) Yes, 40h

(4h + 5)2

Difference of Two Squares A2 – B2 = (A + B)(A – B) This is the reverse from section 9 – 4. (A + B)(A – B) = A2 – B2

Make sure the A and B terms are perfect squares. Ex: a2 – 16 Both are perfect squares a2  a and 16  4. (a + 4)(a – 4)

Ex: m2 – 100 (m + 10)(m – 10)

Are both perfect squares? Yes, 9b2  3b and 25  5 (3b + 5)(3b – 5) Ex: 9b2 - 25 Are both perfect squares? Yes, 9b2  3b and 25  5 (3b + 5)(3b – 5)

Ex: 4w2 - 49 (2w + 7)(2w – 7)

Factor: 14a4 – 14a2 14a2(a2 – 1) 14a2(a + 1)(a - 1) Difference of Squares 14a2(a + 1)(a - 1)

Factor: 3x4 + 30x3 + 75x2 3x2(x2 + 10x + 25)

Factor: 3x4 + 30x3 + 75x2 3x2(x2 + 10x + 25) This is a binomial square.

3x2(x + 5)2

Factor: 2a4 + 14a3 + 24a2 2a2(a2 + 7a + 12) 2a2(a + 3)(a + 4)