Review Polynomials

Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x 2 yw 3, -3, a 2 b 3, and 3yz are all monomials. Polynomials – one or more monomials added or subtracted 4x + 6x 2, 20xy - 4, and 3a 2 - 5a + 4 are all polynomials. Vocabulary

Like Terms Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers. Which terms are like? 3a 2 b, 4ab 2, 3ab, -5ab 2 4ab 2 and -5ab 2 are like. Even though the others have the same variables, the exponents are not the same. 3a 2 b = 3aab, which is different from 4ab 2 = 4abb.

Like Terms Constants are like terms. Which terms are like? 2x, -3, 5b, 0 -3 and 0 are like. Which terms are like? 3x, 2x 2, 4, x 3x and x are like. Which terms are like? 2wx, w, 3x, 4xw 2wx and 4xw are like.

A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Classifying Polynomials PolynomialDegree Classified by degree Classified by number of terms 6 –2 x 3x + 1 –x x – 5 4x 3 – 8x 2 x 4 – 7x 3 – 5x constant linear quartic quadratic cubic monomial binomial polynomial trinomial binomial

Add: (x 2 + 3x + 1) + (4x 2 +5) Step 1: Underline like terms: Step 2: Add the coefficients of like terms, do not change the powers of the variables: Adding Polynomials (x 2 + 3x + 1) + (4x 2 +5) Notice: ‘3x’ doesn’t have a like term. (x 2 + 4x 2 ) + 3x + (1 + 5) 5x 2 + 3x + 6

Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms! Adding Polynomials (x2 (x2 + 3x 3x + 1) + (4x 2 +5) 5x 2 + 3x + 6 (x 2 + 3x + 1) + (4x 2 +5) Stack and add these polynomials: (2a 2 +3ab+4b 2 ) + (7a2+ab+-2b 2 ) (2a 2 +3ab+4b 2 ) + (7a2+ab+-2b 2 ) (2a 2 + 3ab + 4b 2 ) + (7a 2 + ab + -2b 2 ) 9a 2 + 4ab + 2b 2

Adding Polynomials Add the following polynomials; you may stack them if you prefer:

Subtract: (3x 2 + 2x + 7) - (x 2 + x + 4) Subtracting Polynomials Step 1: Change subtraction to addition ( Keep-Change-Change. ). Step 2: Underline OR line up the like terms and add. (3x 2 + 2x + 7) + (- x x + - 4) (3x 2 + 2x + 7) + (- x x + - 4) 2x 2 + x + 3

Subtracting Polynomials Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

1. Add the following polynomials: (9y - 7x + 15a) + (-3y + 8x - 8a) Group your like terms. 9y - 3y - 7x + 8x + 15a - 8a 6y + x + 7a

Combine your like terms. 3a 2 + 3ab + 4ab - b 2 + 6b 2 3a 2 + 7ab + 5b 2 2. Add the following polynomials: (3a 2 + 3ab - b 2 ) + (4ab + 6b 2 )

Add the polynomials. + X2X2 11 X X XY Y Y Y Y 111 X Y Y Y x 2 + 3x + 7y + xy x 2 + 4y + 2x x + 7y x xy + 8

Line up your like terms. 4x 2 - 2xy + 3y 2 +-3x 2 - xy + 2y 2 _________________________ x 2 - 3xy + 5y 2 3. Add the following polynomials using column form: (4x 2 - 2xy + 3y 2 ) + (-3x 2 - xy + 2y 2 )

Rewrite subtraction as adding the opposite. (9y - 7x + 15a) + (+ 3y - 8x + 8a) Group the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a 4. Subtract the following polynomials: (9y - 7x + 15a) - (-3y + 8x - 8a)

Rewrite subtraction as adding the opposite. (7a - 10b) + (- 3a - 4b) Group the like terms. 7a - 3a - 10b - 4b 4a - 14b 5. Subtract the following polynomials: (7a - 10b) - (3a + 4b)

Line up your like terms and add the opposite. 4x 2 - 2xy + 3y 2 + (+ 3x 2 + xy - 2y 2 ) x 2 - xy + y 2 6. Subtract the following polynomials using column form: (4x 2 - 2xy + 3y 2 ) - (-3x 2 - xy + 2y 2 )

Find the sum or difference. (5a – 3b) + (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 3b

Find the sum or difference. (5a – 3b) – (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 9b

Find the sum. Write the answer in standard format. (5x 3 – x + 2 x 2 + 7) + (3x – 4 x) + (4x 2 – 8 – x 3 ) Adding Polynomials SOLUTI ON Vertical format: Write each expression in standard form. Align like terms. 5x x 2 – x + 7 3x 2 – 4 x + 7 – x 3 + 4x 2 – 8 + 4x 3 + 9x 2 – 5x + 6

Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) Adding Polynomials SOLUTI ON Horizontal format: Add like terms. (2 x 2 + x – 5) + (x + x 2 + 6) =(2 x 2 + x 2 ) + (x + x) + (–5 + 6) =3x x + 1

Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) Subtracting Polynomials SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 –2 x 3 + 3x – 4– Add the opposite No change –2 x 3 + 5x 2 – x x 3 – 3x + 4 +

Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) Subtracting Polynomials SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 –2 x 3 + 3x – 4– 5x 2 – 4x + 12 –2 x 3 + 5x 2 – x x 3 – 3x + 4 +

Find the difference. (3x 2 – 5x + 3) – (2 x 2 – x – 4) Subtracting Polynomials SOLUTION Use a horizontal format. (3x 2 – 5x + 3) – (2 x 2 – x – 4)= (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4) = x 2 – 4x + 7 = (3x 2 – 5x + 3) – 2 x 2 + x + 4 = (3x 2 – 2 x 2 ) + (– 5x + x) + (3 + 4)

Multiplying Polynomials Distribute and FOIL

Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2a + 7b)(3a + 5b) Distribute 2a(3a + 5b) and distribute 7b(3a + 5b): 6a ab 21ab + 35b 2 Then add those products, adding like terms: 6a ab + 21ab + 35b 2 = 6a ab + 35b 2

Polynomials * Polynomials An alternative is to stack the polynomials and do long multiplication. (2a + 7b)(3a + 5b) 6a ab 21ab + 35b 2 (2a + 7b) x (3a + 5b) Multiply by 5b, then by 3a: (2a + 7b) x (3a + 5b) When multiplying by 3a, line up the first term under 3a. + Add like terms: 6a ab + 35b 2

Polynomials * Polynomials Multiply the following polynomials:

Polynomials * Polynomials (x + 5) x (2x + -1) -x x x + 2x 2 + 9x + -5 (3w + -2) x (2w + -5) -15w w w + 6w w + 10

Polynomials * Polynomials (2a 2 + a + -1) x (2a 2 + 1) 2a 2 + a a 4 + 2a a 2 + 4a 4 + 2a 3 + a + -1

Types of Polynomials We have names to classify polynomials based on how many terms they have: Monomial: a polynomial with one term Binomial: a polynomial with two terms Trinomial: a polynomial with three terms

F :Multiply the First term in each binomial. 2x 4x = 8x 2 There is an acronym to help us remember how to multiply two binomials without stacking them. F.O.I.L. (2x + -3)(4x + 5) (2x + -3)(4x + 5) = 8x x + -12x = 8x x O :Multiply the Outer terms in the binomials. 2x 5 = 10x I :Multiply the Inner terms in the binomials. -3 4x = -12x L :Multiply the Last term in each binomial = -15

Use the FOIL method to multiply these binomials: F.O.I.L. 1) (3a + 4)(2a + 1) 2) (x + 4)(x - 5) 3) (x + 5)(x - 5) 4) (c - 3)(2c - 5) (2w + 3)(2w - 3)

Use the FOIL method to multiply these binomials: F.O.I.L. 1) (3a + 4)(2a + 1) = 6a 2 + 3a + 8a + 4 = 6a a + 4 2) (x + 4)(x - 5) = x2 x2 + -5x + 4x = x2 x2 + -1x ) (x + 5)(x - 5) = x2 x2 + -5x + 5x = x2 x2 + 4) (c - 3)(2c - 5) = 2c c + -6c + 15 = 2c c ) (2w + 3)(2w - 3) = 4w w + 6w + -9 = 4w

There are three techniques you can use for multiplying polynomials. The best part about it is that they are all the same! Huh? Whaddaya mean? It’s all about how you write it…Here they are! 1)Distributive Property 2)FOIL 3)Box Method Sit back, relax (but make sure to write this down), and I’ll show ya!

1) Multiply. (2x + 3)(5x + 8) Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8). 10x x + 15x + 24 Combine like terms. 10x x + 24 A shortcut of the distributive property is called the FOIL method.

The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. 2) Use the FOIL method to multiply the following binomials: (y + 3)(y + 7).

(y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial. y2y2

(y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial. y 2 + 7y

(y + 3)(y + 7). I tells you to multiply the INNER terms of each binomial. y 2 + 7y + 3y

(y + 3)(y + 7). L tells you to multiply the LAST terms of each binomial. y 2 + 7y + 3y + 21 Combine like terms. y y + 21

Remember, FOIL reminds you to multiply the: F irst terms O uter terms I nner terms L ast terms

The third method is the Box Method. This method works for every problem! Here’s how you do it. Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where. This will be modeled in the next problem along with FOIL. 3x-5 5x +2

3) Multiply (3x - 5)(5x + 2) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 15x x – 10 3x-5 5x +2 15x 2 +6x -25x -10 You have 3 techniques. Pick the one you like the best! 15x 2 +6x -25x -10

4) Multiply (7p - 2)(3p - 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p 2 – 34p + 8 7p-2 3p -4 21p 2 -28p -6p +8 21p 2 -28p -6p +8

Multiply (y + 4)(y – 3) 1.y 2 + y – 12 2.y 2 – y – 12 3.y 2 + 7y – 12 4.y 2 – 7y – 12 5.y 2 + y y 2 – y y 2 + 7y y 2 – 7y + 12

Multiply (2a – 3b)(2a + 4b) 1.4a ab – 12b 2 2.4a 2 – 14ab – 12b 2 3.4a 2 + 8ab – 6ba – 12b 2 4.4a 2 + 2ab – 12b 2 5.4a 2 – 2ab – 12b 2

5) Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x 2 - 5x + 4) - 5(x 2 - 5x + 4) 2x x 2 + 8x - 5x x - 20 Group and combine like terms. 2x x 2 - 5x 2 + 8x + 25x x x x - 20

x2x2 -5x+4 2x -5 5) Multiply (2x - 5)(x 2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. 2x 3 -5x 2 -10x 2 +25x +8x -20 Almost done! Go to the next slide!

x2x2 -5x+4 2x -5 5) Multiply (2x - 5)(x 2 - 5x + 4) Combine like terms! 2x 3 -5x 2 -10x 2 +25x +8x -20 2x 3 – 15x x - 20

Multiply (2p + 1)(p 2 – 3p + 4) 1.2p 3 + 2p 3 + p y 2 – y – 12 3.y 2 + 7y – 12 4.y 2 – 7y – 12

Example: (x – 6)(2x + 1) x(2x)+ x(1)– (6)2x– 6(1) 2x 2 + x – 12x – 6 2x 2 – 11x – 6

2x 2 (3xy + 7x – 2y) 2x 2 (3xy) + 2x 2 (7x) + 2x 2 (–2y) 2x 2 (3xy + 7x – 2y) 6x 3 y + 14x 2 – 4x 2 y

(x + 4)(x – 3) (x + 4)(x – 3) x(x) + x(–3) + 4(x) + 4(–3) x2 x2 – 3x + 4x – 12 x2 x2 + x –

(2y – 3x)(y – 2) (2y – 3x)(y – 2) 2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2) 2y 2 – 4y – 3xy + 6x

There are formulas (shortcuts) that work for certain polynomial multiplication problems. (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 (a - b)(a + b) = a 2 - b 2 Being able to use these formulas will help you in the future when you have to factor. If you do not remember the formulas, you can always multiply using distributive, FOIL, or the box method.

Let’s try one! 1) Multiply: (x + 4) 2 You can multiply this by rewriting this as (x + 4)(x + 4) OR You can use the following rule as a shortcut: (a + b) 2 = a 2 + 2ab + b 2 For comparison, I’ll show you both ways.

1) Multiply (x + 4)(x + 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. x 2 +8x + 16 x+4 x x 2 +4x +16 Now let’s do it with the shortcut! x2x2 +4x +16 Notice you have two of the same answer?

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

2)Multiply: (3x + 2y) 2 using (a + b) 2 = a 2 + 2ab + b 2 (3x + 2y) 2 a = 3x and b = 2y Plug into the formula a 2 + 2ab + b 2 (3x)2 + 2(3x)(2y) + (2y)2 Simplify 9x xy +4y 2

Multiply (2a + 3) 2 1.4a 2 – 9 2.4a a a a a + 9

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

Multiply (x – y) 2 1.x 2 + 2xy + y 2 2.x 2 – 2xy + y 2 3.x 2 + y 2 4.x 2 – y 2

5) Multiply (x – 3)(x + 3) First terms: Outer terms: Inner terms: Last terms: Combine like terms. x 2 – 9 x-3 x +3 x 2 +3x -3x -9 This is called the difference of squares. x2x2 +3x -3x -9 Notice the middle terms eliminate each other!

5) Multiply (x – 3)(x + 3) using (a – b)(a + b) = a 2 – b 2 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 x 2 – 9

6) Multiply: (y – 2)(y + 2) (y) 2 – (2) 2 y 2 – 4 7) Multiply: (5a + 6b)(5a – 6b) (5a) 2 – (6b) 2 25a 2 – 36b 2

Multiply (4m – 3n)(4m + 3n) 1.16m 2 – 9n m 2 + 9n m 2 – 24mn - 9n m mn + 9n 2

Simplify. 1) 2)

Follow the pattern! Last Term Twice the Last Term Square of the Last Term

Difference of Squares. Multiply. 1) 2) 3) 4) Inner and Outer terms cancel!

Multiply. Example 2: Finding Products in the Form (a – b) 2 A. (x – 6) 2 (a – b) = a 2 – 2ab + b 2 (x – 6) = x 2 – 2x(6) + (6) 2 = x – 12x + 36 Use the rule for (a – b) 2. Identify a and b: a = x and b = 6. Simplify. B. (4m – 10) 2

Multiply. Example 2: Finding Products in the Form (a – b) 2 C. (2x – 5y ) 2 D. (7 – r 3 ) 2

Check It Out! Example 2 Multiply. a. (x – 7) 2 b. (3b – 2c) 2

Check It Out! Example 2c Multiply. (a 2 – 4) 2

(a + b)(a – b) = a 2 – b 2 A binomial of the form a 2 – b 2 is called a difference of two squares.

Multiply. Example 3: Finding Products in the Form (a + b)(a – b) A. (x + 4)(x – 4) (a + b)(a – b) = a 2 – b 2 (x + 4)(x – 4) = x 2 – 4 2 = x 2 – 16 Use the rule for (a + b)(a – b). Identify a and b: a = x and b = 4. Simplify. B. (p 2 + 8q)(p 2 – 8q)

Multiply. Example 3: Finding Products in the Form (a + b)(a – b) C. (10 + b)(10 – b)

Check It Out! Example 3 Multiply. a. (x + 8)(x – 8) b. (3 + 2y 2 )(3 – 2y 2 )

Check It Out! Example 3 Multiply. c. (9 + r)(9 – r)