Polynomials and Polynomial Functions

1
Chapter 6

• Polynomials and Polynomial Functions

2
6.1 Power Functions their Inverses

• Power Function y axn
• Even y axis is axis of symmetry
• Odd origin is point of symmetry
• Neither not even or odd
• Extraneous solution solution doesnt work in
  original equation
• Index n given
• Inverse switch x y solve for y
• Rational exponent
• Principal root the positive root

3

• Exponents bn b base n exponent
• x0 1
• x1 x
• Xm xn x mn
• X m / xn x m-n
• (xm)n x mn
• (xayb)n xanybn
• x-m 1/xm ___
• Radicalsradicand, index, power nÖ xp x
  p/n
• Simplify all radicals
• No fractions in radical
• All nth powers are removed from radicand
• Index is as low as possible
• Rationalize all denominators
• Add like with like inside with inside/outside
  with outside
• EIEIO

4
6.1 Power Functions their Inverses

• radical form exponent form
• Square root
• Cube root
• Fourth root
• p.256

5
6.1 Power Functions their Inverses

• Power function Inverse

p.257
6
6.2 Polynomial Functions

• Polynomial function when you add or subtract
  various power functions constants
• Degree the exponent in a term determines the
  degree of that term, the highest exponent in
  the polynomial determines the degree of the
  polynomial
• Standard form terms arranged in descending order
  by degree
• End behavior describes the far left and far
  right portions of the graph for polynomials
  there are 4 types up/up, up/down, down/down,
  down/up

7
6.2 Polynomial Functions

• End behavior
• Even
• odd

8
6.2 Polynomial Functions

• Polynomial Poly.Degree Name using
  degree terms Name usingterms
• 6 0 constant 1 monomial
• 2x 3 1 linear 2 binomial
• 3x² 2 quadratic 1 monomial
• 2x³ - 5x² - 3 3 cubic 3 trinomial

9
6.3 Polynomial and Linear Factors

• Factored form x² (x 5)(x 7) group in ()s
• Simplified form x² 12x 36 no ()s
• Relative maximum in a given range
  the max
• Relative minimum in a given range the
  min
• Zeros from above the x intercepts 0, 0, -5, -7
• Factor theorem if x-b is a factor then b is a
  zero
• Multiple zero when the factor occurs more than
  once
• Multiplicity the number of times the factor (
  and thus the zero) appears

10
6.3 Polynomial and Linear Factors

• Determine zeros their multiplicity
• 0 m. of 2, - m. of 5, -4 m. of 6
• Write each in standard form
• Y (x-2)(x3)
• y x² 3x 2x 6 so y x² x - 6
• Write a polynomial with the zeros
• Given zeros of 3, -1, 0, 0
• (x-3)(x1)x²

11
6.4 Solving Polynomial Equations

• Solve by factoring
• Solve by graphing
• Solve using the quadratic formula
• Note Solve means to find the values of x that
  make the
• equation true.

12
Factoring Review

• GCFalways do this first!!!!
• Simple monomial GCF
• Common binomial factors
• Difference of squares x2 -y2 ? ( x y) (x
  y)
• 32 x2 -98y2
• x22x1 -y24y-4
• Difference/Sum of cubes x3 y3 ? ( x y) (x2
  xy y2)
• x3 y3 ? ( x y) (x2 xy y2)
• 6y3 -48
• Perfect Square Trinomial? SDS (x2 2xy y2)? (x
  y)2
• 16x2 40xy 25y2

13
Factoring continued

• Simple polynomials with quadratic coefficient of
  1
• (x2 4xy -21y2)
• (x3 8x2-42x)
• x2 2 xy y2 -b2
• Polynomials with quadratic coefficient other than
  1
• (2x2 - 7x -15)
• 18d2 19d -12
• 6a2 27a 15 (remember gcf or youll
  reduce it out!)
• Grouping
• x3 2x2 x 2
• 1/16 - 9x2 12xy 4y2
• Combinations of the above

14
6.4 Solving Polynomial Equations after factoring

• X³ 10x² 25x 0
• X (X² - 10X 25) X (X -5)(X 5)
• Solving for x we get x 0 or 5
• (x 5)(x 8)
• This is in factored form
• Solving for x we get x -5 or 8

15
6.4 Solving Polynomial Equations by graphing or
using the quadratic formula

• 12x² 14x 10
• Solve using
• Solve by graphing

16
6.5 Dividing Polynomials
17
6.6 Combinations

• The number of combinations of n objects of a set
  chosen r objects at a time. Order is not
  important.
• nCr n!/(r!(n-r)!)
• How many ways can 5 letters be chosen from the
  alphabet?
• n alphabet 26
• r chosen letters 5
• 26C5 65, 780

18
6.7 Binomial Theorem

• Binomial theorem is used to expand polynomials
  for any binomial
• (ab)n,