POLYNOMIALS Chapter 4

4-1 Exponents

EXPONENTIAL FORM – number written such that it has a base and an exponent 4 3 = 4 4 4

BASE – tells what factor is being multiplied EXPONENT – Tells how many equal factors there are

EXAMPLES 1. x x x x = x = p q 3 p q p = -6p 3 q 2 4. (-2) b (-4) b = 8b 2

ORDER OF OPERATIONS Simplify expression within grouping symbols Simplify powers Simplify products and quotients in order from left to right Simplify sums and differences in order from left to right

EXAMPLES = -(3)(3)(3)(3) = (-3) 4 = (-3)(-3)(-3)(-3) = (1 + 5) 2 = (6) 2 = = = 26

4-2 Adding and Subtracting Polynomials

DEFINITIONS Monomial – an expression that is either a numeral, a variable, or the product of a numeral and one or more variables. Monomial – an expression that is either a numeral, a variable, or the product of a numeral and one or more variables. -6xy, 14, z, 2/3r, ab -6xy, 14, z, 2/3r, ab

DEFINITIONS Polynomial – an expression that is the sum of monomials Polynomial – an expression that is the sum of monomials x + x 2 -4x x + x 2 -4x

DEFINITIONS Binomial – an expression that is the sum of two monomials (has two terms) Binomial – an expression that is the sum of two monomials (has two terms) x, x 2 - 4x x, x 2 - 4x

DEFINITIONS Trinomial – an expression that is the sum of three monomials (has three terms) Trinomial – an expression that is the sum of three monomials (has three terms) x + y, x 2 - 4x x + y, x 2 - 4x + 2

DEFINITIONS Coefficient – the numeral preceding a variable Coefficient – the numeral preceding a variable 2x – coefficient = 2 2x – coefficient = 2

DEFINITIONS Similar terms – two monomials that are exactly alike except for their coefficients Similar terms – two monomials that are exactly alike except for their coefficients 2x, 4x, -6x, 12x, -x 2x, 4x, -6x, 12x, -x

DEFINITIONS Simplest form – when no two terms of a polynomial are similar Simplest form – when no two terms of a polynomial are similar 4x 3 – 10x 2 + 2x - 1 4x 3 – 10x 2 + 2x - 1

DEFINITIONS Degree of a variable– the number of times that the variable occurs as a factor in the monomial Degree of a variable– the number of times that the variable occurs as a factor in the monomial 4x 2 degree of x is 2 4x 2 degree of x is 2

DEFINITIONS Degree of a monomial – the sum of the degrees of its variables. Degree of a monomial – the sum of the degrees of its variables. 4x 2 y degree of monomial is 3 4x 2 y degree of monomial is 3

DEFINITIONS Degree of a polynomial – is the greatest of the degrees of its terms after it has been simplified. Degree of a polynomial – is the greatest of the degrees of its terms after it has been simplified. -6x 3 + 3x 2 + x 2 + 6x 3 – 5 -6x 3 + 3x 2 + x 2 + 6x 3 – 5

Examples (3x 2 y+4xy 2 – y 3 +3) + (x 2 y+3y 3 – 4) (-a 5 – 5ab+4b 2 – 2) – (3a 2 – 2ab – 2b 2 – 7)

4-3 Multiplying Monomials

RULE OF EXPONENTS Product Rule a m a n = a m + n x 3 x 5 = x 8 (3n 2 )(4n 4 ) = 12n 6

4-4 Powers of Monomials

RULE OF EXPONENTS Power of a Power (a m ) n = a mn (x 3 ) 5 = x 15

RULE OF EXPONENTS Power of a Product (ab) m = a m b m (3n 2 ) 3 = 3 3 n 6

4-5 Multiplying Polynomials by Monomials

Examples – Use Distributive Property x(x + 3) x(x + 3) x 2 + 3x 4x(2x – 3) 4x(2x – 3) 8x 2 – 12x -2x(4x 2 – 3x + 5) -2x(4x 2 – 3x + 5) -8x 3 +6x 2 – 10x

4-6 Multiplying Polynomials

Use the Distributive Property (x + 4)(x – 1) (x + 4)(x – 1) (3x – 2)(2x 2 - 5x- 4) (3x – 2)(2x 2 - 5x- 4) (y + 2x)(x 3 – 2y 3 + 3xy 2 + x 2 y) (y + 2x)(x 3 – 2y 3 + 3xy 2 + x 2 y)

4-7 Transforming Formulas

Examples C = 2  r, solve for r c/2  = r

Examples S = v/r, solve for r R = v/s

4-8 Rate-Time- Distance Problems

Example 1 A helicopter leaves Central Airport and flies north at 180 mi/hr. Twenty minutes later a plane leaves the airport and follows the helicopter at 330 mi/h. How long does it take the plane to overtake the helicopter.

Use a Chart RateTimeDistance helicopter180 t + 1/3 180(t + 1/3) plane330t330t

Solution 330t = 180(t + 1/3) 330t = 180t t = 60 t = 2/5

Example 2 Bicyclists Brent and Jane started at noon from points 60 km apart and rode toward each other, meeting at 1:30 PM. Brent’s speed was 4 km/h greater than Jane’s speed. Find their speeds.

Use a Chart RateTimeDistance Brent r (r + 4) Janer1.51.5r

Solution 1.5(r + 4) r = r r = 60 3r + 6 = 60 3r = 54 r = 18

4-9 Area Problems

Examples A rectangle is 5 cm longer than it is wide. If its length and width are both increased by 3 cm, its area is increased by 60 cm 2. Find the dimensions of the original rectangle.

Draw a Picture x + 5 x x + 8 x + 3

Solution x(x+5) + 60 = (x+3)(x + 8) X 2 + 5x + 60 = x 2 +11x = 6x = 6x 6 = x and = 11

Example 2 Hector made a rectangular fish pond surrounded by a brick walk 2 m wide. He had enough bricks for the area of the walk to be 76 m 2. Find the dimensions of the pond if it is twice as long as it is wide.

Draw a Picture 2 m 2x x 2x + 4 x + 4

Solution (2x + 4)(x + 4) – (2x)(x) = 76 2x 2 + 8x + 4x + 16 – 2x 2 = 76 12x + 16 = x = x = 5

4-10 Problems Without Solutions

Examples A lawn is 8 m longer than it is wide. It is surrounded by a flower bed 5 m wide. Find the dimensions of the lawn if the area of the flower bed is 140 m 2

Draw a Picture x + 8 x 5 5

Solution (x+10)(x+18) –x(x+8) = 140 x x –x 2 -8x = x = -40 x = -2 Cannot have a negative width

THE END