Solving Polynomial Functions

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# Solving Polynomial Functions - PowerPoint PPT Presentation

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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### Solving Polynomial Functions

Factoring a Polynomial
• Remember Three Types of Problems
• Difference/Sum of Two Cubes
• Grouping
Practice Problems
• Factor and solve
• x3 – 64
• x3 + 6x2 -4x -24=0
• 4w4 + 40w2- 44=0
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
• Divide f(x)=3x4 – 5x3 + 4x – 6

by (x2 -3x + 5)

x2 -3x + 5 3x4 – 5x3 + 0x2 + 4x – 6

Practice
• Divide f(x)=x3 + 5x2 – 7x + 2 by x-2
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 the polynomial

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

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
• 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

F(x) = x3 + 2x2 – 11x - 12

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

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.

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

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

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 )

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

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.

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