Chapter 4 – Polynomials and Rational Functions

4.1 Polynomial FunctionsDef: A polynomial in one variable, x, is an expression of the form

. The coefficients a0, a1, a2, …, an represent complex numbers (real or imaginary), a0 is not zero, and n represents a nonnegative integer.

Def: The degree of a polynomial in one variable is the greatest exponent of its variable.

Def: If a function f is a polynomial in one variable, then f is a polynomial function.

Def: If p(x) represents a polynomial, then p(x) = 0 is called a polynomial equation.

Def: A root of the equation is a value of x for which the value of the polynomial p(x) is 0. It is also called a zero.

1 20 1 2 1... 0n n

n n na x a x a x a x a

Ex: Determine if each expression is a polynomial in one variable. If so, state the degree.

a. b. c.

Ex: Determine whether 3 is a root of

4 22 4c a c 4 2 5 35 6w w w w 2 33 8x

x

3 22 5 6 0x x x

What is an imaginary number?

What is a complex number?

Fundamental Theorem of AlgebraEvery polynomial equation with degree greater than zero has at least one root in the set of complex numbers

Corollary to the Fundamental Theorem of AlgebraEvery polynomial p(x) of degree n can be written as the product of a constant k and n linear factors.

Thus, a polynomial equation of degree n has exactly n complex roots, namely r1, r2, r3, …, rn.

Relationship with degree and roots:

1 2 3( ) ... np x k x r x r x r x r

Ex: State the number of complex roots of the equation x3 + 2x2 – 8x = 0. Then find the roots and graph the related polynomial function.

Ex: Write the polynomial equation of least degree with roots -3 and 2i.

4.2 Quadratic Equations and Inequalities

2 3 88 0x x 20 10 50 1500x x

Ex: Solve each equation by completing the square.

a. b.

Quadratic FormulaThe roots of a quadratic equation of the form ax2 + bx + c = 0 with a not equal to zero are given by the following formula.

2 4

2

b b acx

a

Ex: Solve 4x2 – 8x + 3 = 0 using the quadratic formula. Then graph the related function.

Discriminantb2 – 4ac > 0 two distinct real roots b2 – 4ac = 0 exactly one real root (double root)b2 – 4ac < 0 no real roots (imaginary roots)

Ex: Determine the discriminant of x2 – 6x + 13 = 0. Use the quadratic formula to find the roots. Then graph the related function.

Ex: Graph y > x2 + 8x - 20

4.3 The Remainder and Factor TheoremsThe Remainder TheoremIf a polynomial p(x) is divided by x – r, the remainder is a constant, p(r), and

where q(x) is a polynomial with degree one less than the degree of p(x).

Example: Let p(x) = x3 + 3x2 – 2x – 8. Show that the value of p(-2) is the remainder when p(x) is divided by x + 2.

( ) ( ) ( ) ( )p x x r q x p r

Ex: Use synthetic division to divide m5 – 3m2 – 20 by m – 2.

The Factor TheoremThe binomial x – r is a factor of the polynomial p(x) if and only if p(r) = 0.

Ex: Let p(x) = x3 – 4x2 – 7x + 10. Determine if x – 5 is a factor of p(x).

4.4 The Rational Root Theorem

Rational Root Theorem:Let represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0.

Example:

Possible values for p:

Possible values for q:

Possible rational roots:

1 20 1 2 1... 0n n

n n na x a x a x a x a

3 20 2 25r r

Integral Root Theorem:Let represent a polynomial equation that has leading coefficients of 1, integral coefficients, and . Any rational roots of this equation must be integral factors of an.

Ex: Find the roots of x3 + 6x2 +10x +3 = 0.

Descartes’ Rule of SignsSuppose p(x) is a polynomial whose terms are arranged in descending powers of the variable. Then the number of positive real zeros of p(x) is the same as the number of changes in sign of the coefficients of the terms, or is less than this by an even number. The number of negative real zeros of p(x) is the same as the number of changes in sign of the coefficients of the terms of p(-x), or is less than this by an even number.

1 20 1 2 1... 0n n

n n na x a x a x a x a

0na

Ex: State the number of possible complex zeros, the number of positive real zeros, and the number of possible negative real zeros for h(x) = x4 – 2x3 + 7x2 + 4x -15.

Ex: Find the zeros of M(x) = x4 +4x3 +3x2 – 4x – 4. Then graph the function.

4.5 Locating the Zeros of a Function

The Location Principle:Suppose y = f(x) represents a polynomial function. If a and b are two numbers with f(a) negative and f(b) positive, the function has at least one real zero between a and b.

Ex: Determine between which consecutive integers the real zeros of f(x) = x3 + 2x2 – 3x -5 are located.

Ex: Approximate to the nearest tenth the real zeros of f(x) = x4 – 3x3 – 2x2 + 3x – 5. Then sketch the graph of the function, given that the relative maximum is at (0.4, -4.3) and the relative minima are at (-0.7, -6.8) and (2.5, -17.8).

Upper Bound TheoremSuppose c is a positive integer and p(x) is divided by x – c. If the resulting quotient and remainder have no change in sign, then p(x) has no real zeros greater than c. Thus, c is an upper bound of zeros of p(x).

Ex: Find a lower bound of the zeros of f(x) = x4 – 3x3 – 2x2 +3x – 5.

4.6 Rational Equations and Partial Fractions

Ex: Solve

Ex: Solve

2

4 3 16

4 4

t

t t t t

110 20

h h

Decompose into partial fractions

Solve

2

11

2 3

x

x x

2

3 40

5 6

x x

x x

Solve:1 5 1

4 8 2a a

4.7 Radical Equations and InequalitiesSolve:

Solve:

5 4 2x

33 4 12x

Ex: Solve

Ex: Solve

3 4 2 7 3x x

5 4 8x