Polynomials. DegreeNameExample 0Constant 1Linear 2Quadratic 3Cubic 4Quartic 5Quintic Some of the Special Names of the Polynomials of the first few degrees:

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Presentation transcript:

Polynomials

DegreeNameExample 0Constant 1Linear 2Quadratic 3Cubic 4Quartic 5Quintic Some of the Special Names of the Polynomials of the first few degrees:

A polynomial with one term is called a monomial. Example: A polynomial with two terms is called a binomial. Example: A polynomial with three terms is called a trinomial. Example:

Find a quartic function with zeros and and Find the zeros of the function: Answers: Answer:

Find a cubic function with zeros 2 + i and 1. Answer: Find a quartic function with zeros and Answer:

If 4 + i is a root of the polynomial equation below, find the other roots. Answer: If the polynomial function below is divisible by z – 3 – i, find the other roots. Answer:

If is a zero of the function find the value of a. Answer: a = -12 Find the equation of a cubic function with x-intercepts at 1, 2 and 3 and a y-intercept at -6. Answer:

Use synthetic division to divide: by Answer: Use synthetic division to divide: by Answer:

The Remainder Theorem: When a polynomial P(x) is divided by x – a, the remainder is P(a).P(a). Example: Find the remainder when the function below is divided by x – 1. Answer: Remainder is -1

The polynomial below has a remainder of 4 when divided by (x + 1). Calculate the value of k. Answer: k = 3 The polynomial below has a remainder of 8 when divided by (x - 1). Calculate the value of a. Answer: a = 4

The Factor Theorem For a polynomial P(x), x – a is a factor if and only ifP(a) = 0. Example: Below is a polynomial equation and one of its roots, find the remaining roots. Answer: Remaining roots are:

Example: Below is a polynomial equation and one of its roots, find the remaining roots. Answer: Remaining roots are: Example: Below is a polynomial equation and one of its roots, find the remaining roots. Answer: Remaining roots are:

The polynomial below has a factor of (x – 1) and a remainder of 8 when divided by (x +1). Calculate the values of a and b. Answer: a = -5, b = 6 The polynomial is a factor of the polynomial below. Calculate the value of a. Answer: a = 1

Solve the following polynomials: Answer:

Solve the following polynomials: Answer:

Rational Root Theorem Let P (x) be a polynomial of degree n with integral coefficients and a non-zero constant term: If one of the roots of the equation P(x) = 0 is where p and q are non-zero integers with no common factor other than 1, Then p must be a factor of, and q must be a factor of.

Given: Find all of the roots using the rational zero test. Given: Find all of the roots using the rational zero test.