2.4 – Real Zeros of Polynomial Functions

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2.4 – Real Zeros of Polynomial Functions

By the end of Monday, you will be able to…..
Use Long division and Synthetic Division to divide polynomials Apply Remainder Theorem and Factor Theorem Find the upper and lower bounds for zeros of polynomial functions Find the real zeros of a polynomial function

Recall: Division Terminology
Dividend Divisor Quotient Remainder

Long Division Example

You Try! Long Division

Division Algorithm for Polynomials
f(x) = d(x) q(x) + r(x) f(x) – polynomial (dividend) d(x) – polynomial (divisor) q(x) – unique polynomial (quotient) r(x) – unique polynomial (remainder) Note: r(x) = 0 or the degree of r is less than the degree of d.

Long Division f(x) = x2 – 2x + 3 d(x) = x – 1

Long Division 2) f(x) = x4 – 2x3 + 3x2 – 4x + 6 d(x) = x2 + 2x - 1

Long Division- You Try! f(x) = x3 + 4x2 + 7x – 9 d(x) = x + 3

Remainder and Factors Theorem
Remainder Theorem - If a polynomial f(x) is divided by x - k, then the remainder is r = f(k). Note: So if you want to know the remainder after dividing by x-k you don't need to do any division: Just calculate f(k). Factor Theorem - A polynomial function f(x) has a factor x - k if an only if f(k) = 0.

1) f(x) = x3 – x2 + 2x – 1 k = -3 2) 2x3 – 3x2 + 4x – 7 k = 2
Use the Remainder Theorem to find the remainder when f(x) is divided by x-k 1) f(x) = x3 – x2 + 2x – 1 k = -3 2) 2x3 – 3x2 + 4x – 7 k = 2

You Try! Use the Remainder Theorem to find the remainder when f(x) is divided by x-k
Ex1) f(x) = 2x2 – 3x + 1 k = 2 Ex2) f(x) = 3x4 + 2x3 + 4x k = -5

Important Connections for Polynomial Functions
The following statements are all equivalent (for a polynomial function f and a real number k): x = k is a solution (or root) of the equation f(x) = 0. k is a zero of the function f. k is an x-intercept of the graph of y = f(x). x - k is a factor of f(x).

Let’s take a look at Synthetic Division!
Ex) f(x) = x3 – 5x2 + 3x -2 d(x) = x+1

You try Synthetic Division:
Ex) f(x) = 9x3 + 7x2 – 3x d(x) = x - 10

More Synthetic Division
Ex) f(x) = 5x4 – 3x + 1 d(x) = 4 - x

Upper and Lower Bound Tests for Real Zeros
Suppose f(x) is divided by (x – k) (use synthetic division): If k > 0 and every number in the last line is positive or zero, then k is an upper bound for the real zeros of f. If k < 0 and the numbers in the last line are alternately non-negative and non-positive, then k is a lower bound for the real zeros of f.

Use synthetic division to prove that the number k is an upper bound for the real zeros of the function f Ex) k = 3 f(x) = 4x4 – 35x2 - 9

Use synthetic division to prove that the number k is a lower bound for the real zeros of the function f Ex) k = 0 f(x) = x3 – 4x2 + 7x -2

You Try! Use synthetic division to prove that the number k is an upper bound for the real zeros of the function f Ex) k = 3 f(x) = 2x3 – 4x2 + x - 2 Use synthetic division to prove that the number k is a lower bound for the real zeros of the function f Ex) k = -1 f(x) = 3x3 – 4x2 + x +3

Establishing bounds for real zeros
Ex) Show that all the zeros of f(x) = 2x3 – 3x2 – 4x + 6 lie within the interval [-7,7].

Rational Zeros (roots) Theorem
If a polynomial has any rational roots, then they are in the form of p q p is a factor of the constant term q is a factor of the leading coefficient

Example Using the Rational Zeros Theorem
List all the possible rational roots of f(x) = 2x3 – 3x2 – 4x + 6 We found the possible rational roots, but which ones are actually the roots?

f(x) = x3 + x2 – 10x + 8 Let’s try another one: Reminders:
Find the possible rational roots Look at the graph to see which roots to test Test roots using synthetic division If the remainder is 0, then it is indeed a root If not, then test another possible and reasonable root Look at factors of function and factor to find the rest of the roots

You try! Find all of the real zeros of the function.
f(x) = x3 + x2 – 8x - 6 Reminders: Find the possible rational roots Look at the graph to see which roots to test Test roots using synthetic division If the remainder is 0, then it is indeed a root If not, then test another possible and reasonable root Look at factors of function and factor to find the rest of the roots. If you can’t factor, use the quadratic formula!