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 TerminologyDividendDivisorQuotientRemainder

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 Division1) f(x) = x2 – 2x + 3 d(x) = x – 1

Long Division

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

Long Division- You Try!1) f(x) = x3 + 4x2 + 7x – 9

d(x) = x + 3

Remainder and Factors TheoremRemainder 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.

Use the Remainder Theorem to find the remainder when f(x) is divided by x-k1) f(x) = x3 – x2 + 2x – 1

k = -3

2) 2x3 – 3x2 + 4x – 7k = 2

You Try! Use the Remainder Theorem to find the remainder when f(x) is divided by x-kEx1) 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):1. x = k is a solution (or root) of

the equation f(x) = 0.2. k is a zero of the function f.3. k is an x-intercept of the graph

of y = f(x).4. 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 DivisionEx) f(x) = 5x4 – 3x + 1

d(x) = 4 - x

Upper and Lower Bound Tests for Real ZerosSuppose 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 fEx) 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 fEx) 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 = 3f(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 zerosEx) Show that all the zeros of f(x) = 2x3 – 3x2 – 4x + 6 lie within

the interval [-7,7].

Rational Zeros (roots) TheoremIf 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?

Let’s try another one:

f(x) = x3 + x2 – 10x + 8

Reminders:

1)Find the possible rational roots2)Look at the graph to see which roots to

test3)Test roots using synthetic division4)If the remainder is 0, then it is indeed a

root5)If not, then test another possible and

reasonable root6)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:

1)Find the possible rational roots2)Look at the graph to see which roots to

test3)Test roots using synthetic division4)If the remainder is 0, then it is indeed a

root5)If not, then test another possible and

reasonable root6)Look at factors of function and factor to

find the rest of the roots. If you can’t factor, use the quadratic formula!

Don’t forget your homework!

Pg. 216-218 (4-60 every 4, 26, 58)