Transcript Section 2.4: Real Zeros of Polynomial Functions
April 7, 2015
Polynomial Long Division
Divide by w – 3. 3
x
3 2
x
2 4
x
3
x
3
x
3
Synthetic Division
3
x
3 6
x
2
Remainder Theorem
If dividing f(x) by x – a, f(a) will determine the remainder.
x
7
x
6
x – 4?
Use the Remainder Theorem to evaluate f (x) = 6x 3 5x 2 + 4x – 17 at x = 3.
–
Factor Theorem
When dividing polynomials, if the remainder is zero, then the divisor is a factor.
Use the Factor Theorem to determine whether x – 1 is a factor of f (x) = 2x 4 + 3x 2 – 5x + 7.
Using the Factor Theorem, verify that x + 4 is a factor of f (x) = 5x 4 + 16x 3 – 15x 2 + 8x + 16.
Finding Exact Irrational Zeros
Find the exact zeros for the function. Identify each zero as rational or irrational.
Writing Functions w/ Given Conditions
If the zeros of the function are -3, 7, and -1, and the leading coefficient is 4, write an equation for the function.
Rational Zeros Theorem
Given a function with constant of p and leading coefficient of q, all possible rational zeros can be found by
factors of p factors of q
2
x
3 7
x
2 8
x
6 Find the possible rational zeros of 3
x
3 5
x
2 9
x
16
Upper and Lower Bounds Test
When dividing polynomials, if the quotient polynomial has all non-negative coefficients, then the “k” value is an upper bound. If the quotient polynomial has alternating sign coefficients, then the “k” value is a lower bound.
Show that all real roots of the equation
x
3 10
x
2 12
x
20 lie between - 4 and 4.