9.1 adding and subtracting polynomials. monomial is an expression that is a number, variable, or a...

9.1 Adding and Subtracting Polynomials

MonomialIs 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.

a)18 degree ___

The degree of a nonzero constant is __.

Ex 1: Find the degree of the monomial.

a)18 degree _0_

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_.

PolynomialIs 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.

BinomialIs a polynomial of two terms.

Ex: 3c + 4

TrinomialIs a polynomial of three terms.

Ex: 2x4 + x2 - 2

Ex:

(5x2 + 3x +4) + (3x2 + 5)

8x2 + 3x +9

Adding Horizontally

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

01

0

33

33

xx

xx

•Is x-3 the additive inverse of –x3?

NO

01

0)(

33

33

xx

xx

•Is x-3 the additive inverse of –x-3?

•Yes

011

0)(

33

33

xx

xx

Ex 1: Find the Additive Inverse

7x4 - 3x + 5-7x4 + 3x – 5

Notice the signs changed

Subtract(5x2 + 3x – 2) – (2x2 + 1)

Change the sign of the terms in the 2nd

parenthesis.

Subtract(5x2 + 3x – 2) – (2x2 + 1)

5x2 + 3x – 2 – 2x2 – 1Subtract 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

3x2 + 3x - 3

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)

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 • x10x2 = 2• x • x • x6x = 2 • 3 • xWhich prime factors are in term?

2x4 = 2• x • x • x • x10x2 = 2• x • x • x6x = 2 • 3 • xWhich 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

16x = 2 • • 2 • • x

List the prime factors.

4x3 = 2• 2 •x• x • x •

12x2 = 2 • 2 • 3 • x • x

16x = 2 • 2 • 2 • 2 • x

List the prime factors.

4x3 = 2• 2 •x• x • x •

12x2 = 2 • 2 • 3 • x • x

16x = 2 • 2 • 2 • 2 • x

The GCF is 4x.

Now Factor Out the GFC.

4x3 +12x2 – 16x

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

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

( ) (x2 + - )

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

Ex: 5x3 + 105 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)

Ex: 18y4 – 6y3 + 12y2 6y2 goes into each term.

6y2(3y2) - 6y2(y) + 6y2(2)

6y2(3y2 – y + 2)

Ex: 8x4y3 + 6x2y4 2x2y3 goes into each term.

2x2y3(4x2) + 2x2y3(3y)

2x2y3(4x2 + 3y)

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)

(x + 5)(x + 4)

x2

First terms

(x + 5)(x + 4)

4x

Outside terms

(x + 5)(x + 4)

5x

Inside terms

(x + 5)(x + 4)

20

Last terms

x2 + 4x + 5x + 20

x2 + 9x + 20

Ex:(x + 2)(x + 3)

x2 + 3x + 2x + 6x2 + 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?

NO

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)

5) Multiply (x – 3)(x + 3)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

x2 – 9

x -3

x

+3

x2

+3x

-3x

-9

This is called the difference of squares.

x2

+3x-3x-9

Notice the middle terms

eliminate each other!

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)(y)2 – (2)2

y2 – 4

Ex: Multiply: (5a + 6b)(5a – 6b)

(5a)2 – (6b)2

25a2 – 36b2

Multiply (4m – 3n)(4m + 3n)

1. 16m2 – 9n2

2. 16m2 + 9n2

3. 16m2 – 24mn - 9n2

4. 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 Mistake25)5( 22 xx

Common Mistake

2949

2547

252)52(

2Let x

25)5(

2

22

22

xx

Ex:1) Multiply (x + 4)(x + 4)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

x2 +8x + 16

x +4

x

+4

x2

+4x

+4x

+16

Now let’s do it with the shortcut!

x2

+4x+4x+16

Notice you have two

of the same

answer?

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 = 4Plug into the formula

a2 + 2ab + b2

(x)2 + 2(x)(4) + (4)2

Simplify.x2 + 8x+ 16

This is the same answer!

That’s why the 2 is in the formula!

Ex: 2) Multiply: (3x + 2y)2

using (a + b)2 = a2 + 2ab + b2

(3x + 2y)2

a = 3x and b = 2y

Plug into the formulaa2 + 2ab + b2

(3x)2 + 2(3x)(2y) + (2y)2Simplify

9x2 + 12xy +4y2

Multiply (2a + 3)2

1. 4a2 – 9

2. 4a2 + 9

3. 4a2 + 36a + 9

4. 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

1. x2 + 2xy + y2

2. x2 – 2xy + y2

3. x2 + y2

4. x2 – y2

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

To factor a trinomial of in the form of

x2 + bx + cWhere c > 0

We must find 2 numbers that:

A.) their product is c andB.) 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 61 and 6

2 and 3

Factors of 61 and 6 sum is 7

2 and 3sum 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

Factor x2 + 8x + 12

b = 8, c = 12

What factors of 12 add up to 8?

Factors of 121 and 12 -1 and -12

2 and 6 -2 and -6

3 and 4 -3 and -4

2 and 6

(x + 2)(x + 6)

Factorx2 – 10x + 16 b = -10, c = 16

What factors of 16 add up to -10?

Factors of 161 and 16 -1 and 16

2 and 8 -2 and -8

4 and 4 -4 and -4

-2 and -8

(x – 2)(x – 8)

Factorp2 – 3pq + 2q2

b = -3, c = 2

What factors of 2 add up to -3?

1 and 2 -1 and -2

-1 and -2

(p – q)(p – 2q)

Factors of -2

To factor a trinomial of in the form of

x2 + bx + cWhere 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 = -10What factors of -10 have the sum of -3.

Factors of -101 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 = -4What factors of -4 have the sum of 3.

Factors of -41 and -4 -1 and 4

2 and -2

-1 and 4

(x – 1)(x + 4)

Factor y2 – 12yz – 28z2

b = -12 and c = -28What factors of -28 have the sum of -12.

Factors of -281 and -28 -1 and 28

2 and -14 -2 and 14

4 and -7 -4 and 7

2 and -14

(x + 2z)(x – 14z)

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

1) Factor. y2 + 6y + 8Put the first and last terms into the

box as shown.

What are the factors of y2?

y and y

y2

+ 8

1) Factor. y2 + 6y + 8Place the factors outside the box as

shown.

y2

+ 8

y

y

What are the factors of + 8?

+1 and +8, -1 and -8

+2 and +4, -2 and -4

The second box works. Write the numbers on the outside of box for your solution.

1) Factor. y2 + 6y + 8Which box has a sum of + 6y?

y2

+ 8

y

y

y2

+ 8

y

y+ 1 + 2

+ 8 + 4

+ y

+ 8y + 4y

+ 2y

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.

x2

- 63

2) Factor. x2 - 2x - 63Put the first and last terms into the box

as shown.

What are the factors of x2?

x and x

2) Factor. x2 - 2x - 63 Place the factors outside the box as

shown.

x2

- 63

x

x

What are the factors of - 63?

Remember the signs will be different!

2) Factor. x2 - 2x - 63Use trial and error to find the correct

combination!

Do any of these combinations work?

The second one has the wrong sign!

x2

- 63

x

x

+ 21

- 3

+21x

-3x x2

- 63

x

x - 7

+ 9

-7x

+9x

2) Factor. x2 - 2x - 63Change the signs of the factors!

Write your solution.

(x + 7)(x - 9)

x2

- 63

x

x + 7

- 9

+7x

-9x

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

Ex 3: m2 + 6mn + 9n2

Perfect squares?

m2 m and 9n2 3n

(m + 3n)2

Ex 2: if A ≠ 1

64y2 + 48y + 9

Are 64y2 and 9 perfect squares?

YES

(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

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)

Ex: 9b2 - 25Are 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)

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)