Polynomials

Integrated Math 4

Mrs. Tyrpak

Definition

Let n be a nonnegative integer and let be real numbers and exponents be positive.

This is called a polynomial function of degree n.

Key Terms

Let

Let’s label the degree, leading coefficient, and constant term. (MP)

You try one…

Find a partner.

If you’re the tallest, say the degree.

If you’re the shortest, say the constant term.

Say the leading coefficient at the same time.

Relationships?

The Leading Coefficient Test

L.C.EVEN

ODD

Same

Different

L.C.

DEGREE

Real Zeros

The following are equivalent statements about real zeros of a polynomial function:

1. x = a is a zero (solution or root) of the function

2. (a, 0) is an x-intercept of the graph

3. (x – a) is a factor of the polynomial

Finding All Zeros

First we are going to use the rational zeros theorem: If a polynomial has integer coefficients, every rational zero has the form , where p is a factor of the constant term, and q is a factor of the leading coefficient.

Rational Zeros Theorem

Possible rational zeros:

Synthetic Division

Second, we are going to use synthetic division to test each possible zero.

The Remainder and Factor Theorems

• Remainder Theorem: If a polynomial f(x) is divided by x – k, the remainder is r = f(k).

• Factor Theorem: A polynomial f(x) has a factor (x - k) if and only if f(k)=0.

Factor the Polynomial

Thirdly, we are going to factor using the remainder and factor theorems.

Find all the zeros

• Lastly, we will solve each factor for each zero.

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, where n > 0, f has at least one zero in

the complex number system.

Multiplicity

If is a factor of a polynomial …

k is odd, then the graph crosses the x-axis at (c, 0)

k is even, then the graph is tangent to the x-axis at (c,0)

𝑓 (𝑥 )=(𝑥+5 )3 (𝑥−4 ) (𝑥+1 )2

Find the zeros of the polynomial function and state the multiplicity of each and what happens at each zero.

Finding All Zeros and Factors

• Find all the zeros of

Take 5 minutes and try this one..

𝑔 (𝑥 )=𝑥4+6 𝑥3+10 𝑥2+6 𝑥+9

Let’s try and work backwards

• Find a polynomial with integer coefficients that has the given zeros:

1, 5i, -5i

Thanks for your attention!

Don’t forget to complete both the extension and enrichment assignments

before you move on.

Keep up the hard work!