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Solving Polynomial Functions. Factoring a Polynomial. Remember Three Types of Problems Difference/Sum of Two Cubes Grouping Quadratic Like. Practice Problems. Factor and solve x 3 – 64 x 3 + 6x 2 -4x -24=0 4w 4 + 40w 2 - 44=0. Using Division to Find a Zero.

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Presentation Transcript
factoring a polynomial
Factoring a Polynomial
  • Remember Three Types of Problems
    • Difference/Sum of Two Cubes
    • Grouping
    • Quadratic Like
practice problems
Practice Problems
  • Factor and solve
  • x3 – 64
  • x3 + 6x2 -4x -24=0
  • 4w4 + 40w2- 44=0
using division to find a zero
Using Division to Find a Zero
  • If you know one zero, you can use division to find another.
  • Remainder Theorem- If a polynomial f(x) is divided by (x-k) then the remainder is r=f(x)
  • Two forms of polynomial division: long division and synthetic division
long division can be used on any polynomial
Long Division- can be used on any polynomial
  • Divide f(x)=3x4 – 5x3 + 4x – 6

by (x2 -3x + 5)

x2 -3x + 5 3x4 – 5x3 + 0x2 + 4x – 6

practice
Practice
  • Divide f(x)=x3 + 5x2 – 7x + 2 by x-2
synthetic division
Synthetic Division
  • Can be used to divide any polynomial by a divisor of the form x – k
  • To set up synthetic division, list the coefficients in a row.
factor theorem a polynomial f x has a factor x k if f k 0
Factor Theorem- a polynomial f(x) has a factor x-k if f(k)=0

Factor the polynomial

F(x) = 3x3 – 4x2 – 28x -16 completely given that x+ 2 if a factor.

slide10
The profit P ( in millions of dollars) for a shoe manufacturer can be modeled by P= -21x3 + 46x where x is the number of shoes produced (in millions). The company produces 1 million shoes and makes a profit of 25,000,000 but would like to cut back on production. What lesser number of shoes could the company make and still make the same profit?
rational zero theorem
Rational Zero Theorem
  • One way to narrow down the possible zeros of a function is to use the Rational Zero Theorem.
  • If f(x)= anxn + …anx + a0 has integer coefficients, then every rational zero of f has the following form

p = factors of constant term a0

q factors of leading coefficient an

slide14
Find all real zeros of

f(x)= 10x4 - 11x3 – 42x2+ 7x + 12

Step 1: List possible rational zeros.

Step 2: Use graphing calculator to narrow down choice\

Step 3: Use synthetic division to test zero

fundamental theorem of algebra
Fundamental Theorem of Algebra

Theorem: If f(x) is a polynomial of degree n where n >0 then the equation f(x)=0 has at least one solution in the set of complex numbers.

Corollary: if f(x) is a polynomial of degree n where n>0 then the equation f(x)=0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions and so on.

slide17
Complex Conjugate Theorem
  • If f is a polynomial function with real coefficients and a + bi is an imaginary zero of f, then a- bi is a zero of f.

Irrational Conjugate Theorem

Suppose f is a polynomial function with rational coefficients and a and b are rational numbers such that √ b is irrational. If a + √b is a zero of f , then a – √b is also a zero of f.

find the zeros
Find the zeros

F(x) = x5 – 4x4 + 4x3 + 10x2 – 13x - 14

slide19
Write a polynomial function f of least degree that has rational coefficients , a leading coefficient of 1 and 3 and 2 + √5 as zeros.

Set up factors

F(x)= (x – 3)(x – (2 + √5 ))(x – (2 - √5 )

slide20
Write the polynomial function f of least degree that has rational coefficients a leading coefficient of 1 and the given zeros
  • -1, 2, 4

2. 3, 3-i

descartes rule of signs
Descartes Rule of Signs

Let f(x) = anxn + …anx + a0 be a polynomial function with real coefficients.

-The number of positive real zeros of f is equal to the number of changes in sign of coefficients of f(x) or is less than this by an even number.

- The number of negative real zeros of f is equal to the number of changes in sign of coefficients of f(-x) or is less than this by an even number.

slide22
Determine the possible numbers of positive real zeros, negative real and imaginary zeros for the function

F(x)= x6 – 2x5 + 3x4 – 10x3 – 6x2 -8x -8