Chapter 4, 5, and 13 Notes

4-7 Exponents and Multiplication ZTo multiply numbers or variables with the same base, add the exponents. ZExamples: = = = = = =2 5

You can also do the same thing with variables a 5 a 1 b 2 m 5 m 7 y 2 y 3 z z 4

You can also do the same thing with variables - answers a 5 a 1 b 2 = a 5+1 b 2 = a 6 b 2 m 5 m 7 = m 5+7 = m 12 y 2 y 3 z z 4 = y 2+3 z 1+4 = y 5 z 5

Simplify: 1.-2x 2 3x 5 2.6a 3 3a 3.-5c 2 -3c 7

Simplify: Answers 1.-2x 2 3x 5 = -2 3 x 2 x 5 = -6x 7 2.6a 3 3a = 18a c 2 -3c 7 = 15c 9

Finding a Power of a Power ZTo find a power of a power, multiply the exponents ZExamples: 1.(3 2 ) 3 2.(2 4 ) 2 3.(7 2 ) 3

Finding a Power of a Power - Answers ZTo find a power of a power, multiply the exponents ZExamples: 1.(3 2 ) 3 = (3) 2 3 = 3 6 = (2 4 ) 2 = (2) 4 2 = 2 8 = (7 2 ) 3 = (7) 2 3 = 7 6

ZSimplify each expression: 1.(c 5 ) 4 2.(m 3 ) 2 3.(3x 2 ) 2

ZSimplify each expression: Answers 1.(c 5 ) 4 = (c) 5 4 = c 20 2.(m 3 ) 2 = (m) 3 2 = m 6 3.(3x 2 ) 2 = 9x 4

4-8 Exponents and Division ZTo divide numbers or variables with the same base, subtract the exponents Zx 0 = 1 ZExamples:

4-8 Exponents and Division - Answers ZTo divide numbers or variables with the same base, subtract the exponents Zx 0 = 1 ZExamples: = = 3 3 = = = 10 3 = 1,

Simplify each expression 1.X 25 x m 5 3m

Simplify each expression 1.x 25 = x = x 7 x m 5 = 4m 5-1 = 4m 4 3m

Zero as an Exponent 1. (-8) 2 (-8) x 6 5x 6

Zero as an Exponent - Answers 1. (-8) 2 = (-8) 2-2 = (-8) 0 = 1 (-8) x 6 = 25x 6 = 5x 6-6 = 5 5x 6

Negative/Positive Exponents y 8 9y 12

Negative/Positive Exponents - Answers = = 5 -2 = 1 = y 8 = 1y 8 = 1 9y 12 3y 12 3y 4

Using exponents without a fraction bar 1.x 2 y 3 x 3 y 2.m 3 n 2 m 6 n 8

Using exponents without a fraction bar - Answers 1.x 2 y 3 = x 2-3 y 3-1 = x -1 y 2 x 3 y 2.m 3 n 2 = m 3-6 n 2-8 = m -3 n -6 m 6 n 8

5-9 Powers of Products and Quotients ZTo raise a product to a power, raise each factor to the power. Examples: 1.(4x 2 ) 3 2.(xy 2 ) 5

5-9 Powers of Products and Quotients - Answers ZTo raise a product to a power, raise each factor to the power. Examples: 1.(4x 2 ) 3 = (4) 3 (x 2 ) 3 = 64x 6 2.(xy 2 ) 5 = x 5 y 10

Working with a negative sign 1.(-5x) 2 2.-(5x) 2 3.(-5a 2 b 3 ) 2 4.-(2y) 4

Working with a negative sign - Answers 1.(-5x) 2 = (-5) 2 x 2 = 25x 2 2.-(5x) 2 = -(5 2 )(x 2 ) = -25x 2 3.(-5a 2 b 3 ) 2 = 25a 4 b 3 4.-(2y) 4 = -16y 4

Powers of Quotients ZTo raise a quotient to a power, raise both the numerator and denominator to the power. 1.(3/b) 2 2.(2x 2 /3) 3

Powers of Quotients - Answers ZTo raise a quotient to a power, raise both the numerator and denominator to the power. 1.(3/b) 2 = 3 2 /b 2 = 9/b 2 2.(2x 2 /3) 3 = 8x 6 /27

13-4 Polynomials ZPolynomial - a monomial or a sum or difference of monomials ZMonomial - a real number, a variable, or a product of a real number and variables

Is the expression a monomial? Explain Z7x 2 y Z8 + a Za/7y Z5x/4

Is the expression a monomial? Explain -Answers Z7x 2 y - yes, it is the product of 7 and the variables x and y Z8 + a - no, the expression is a sum Za/7y - no, the denominator contains a variable Z5x/4 - yes, it is the product of 5/4 and the variable x

Polynomial# of termsExamples Monomial 14, 32, x, x 2 Binomial 2x-3, 5x+1, x 3 -x Trinomial 3x 2 + x + 1 x 4 -2x -5

Naming a Polynomial Zx - y Z8xyz Zy 2 + 8y + 18 Z10

Naming a Polynomial - Answers Zx - y binomial Z8xyz monomial Zy 2 + 8y trinomial Z monomial

Evaluating a Polynomial ZSubstitute values for the variables m = 8 and p = mp 2.3m - 2p 3. m 2 + 2p - 3

Evaluating a Polynomial - Answers ZSubstitute values for the variables m = 8 and p = mp = m - 2p = m 2 + 2p - 3 = 55

13-5 Add/Subtract Polynomials ZAdd by combining like terms or aligning like terms Example: (2x 2 + 3x -1) + (x 2 + x -3)= (2x 2 + x 2 ) + (3x + x ) + ( ) = 3x 2 + 4x -4

More Examples 1.(z 2 + 5z + 4) + (2z 2 -5) 2.(4x + 9y) +(3x - 5y) 3.a 2 + 6a a 2 - 8a

More Examples - Answers 1.(z 2 + 5z + 4) + (2z 2 -5) = 3z 2 + 5z -1 2.(4x + 9y) +(3x - 5y) 7x + 4y 3.a 2 + 6a a 2 - 8a 9a 2 - 2a - 4

Subtracting Polynomials ZTo subtract polynomials, you have to add the opposite of each term in the 2nd polynomial. ZExample: (5x x) - (3x - 12) 5x x + - 3x x 2 + 7x + 12

More examples 1.(7a 2 - 2a) - (5a 2 + 3a) 2.(10z 2 + 6z + 5) - (z 2 -8z + 7) 3.(3w v) - (5w v)

More examples - Answers 1.(7a 2 - 2a) - (5a 2 + 3a) = 2a 2 - 5a 2.(10z 2 + 6z + 5) - (z 2 -8z + 7) = 9z z (3w v) - (5w v) = -2w v

13-6 Multiplying a Polynomial by a Monomial ZUse the distributive property to multiply each term of the polynomial by the monomial. ZExample: 2x(x + 4) = 2x(x) + 2x(4) = 2x 2 + 8x

More examples 1.3x 2 (8x 2 - 5x + 2) 2.2b(b 2 + 3b -6) 3.3x (2x 2 + x + 3) 4.x(x 2 + 2x + 4)

More examples - Answers 1.3x 2 (8x 2 - 5x + 2) = 24x x 3 + 6x 2 2.2b(b 2 + 3b -6) = 2b 3 + 6b b 3.3x (2x 2 + x + 3) = 6x 3 + 3x 2 + 9x 4.x(x 2 + 2x + 4) = x 3 + 2x 2 + 4x