5.3

Polynomials and Polynomial Functions

Polynomial Vocabulary

Term – a number or a product of a number and variables raised to powers

Coefficient – numerical factor of a term

Constant – term which is only a number

Polynomial – a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

In the polynomial 7x5 + x2y2 – 4xy + 7

There are 4 terms: 7x5, x2y2, -4xy and 7.

The coefficient of term 7x5 is 7,

of term x2y2 is 1,

of term –4xy is –4 and

of term 7 is 7.

7 is a constant term.

Polynomial Vocabulary

Monomial is a polynomial with one term.

Binomial is a polynomial with two terms.

Trinomial is a polynomial with three terms.

Types of Polynomials

Degree of a term

The degree of a term is the sum of the exponents on the variables contained in the term.

Degree of a constant is 0.

Degree of the term 5a4b3c is 8 (remember that c can be written as c1).

Degrees

Degree of a polynomial

The degree of a polynomial is the greatest degree of all its terms.

Degree of 9x3 – 4x2 + 7 is 3.

Degrees

Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved.

Find the value of 2x3 – 3x + 4 when x = 2.

= 2(2)3 – 3(2) + 42x3 – 3x + 4

= 2(8) + 6 + 4

= 6

Evaluating Polynomials

Example:

Like terms are terms that contain exactly the same variables raised to exactly the same powers.

Combine like terms to simplify.

x2y + xy – y + 10x2y – 2y + xy

Only like terms can be combined through addition and subtraction.

Warning!

11x2y + 2xy – 3y= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y =

= x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together)

Combining Like Terms

Example:

To Add Polynomials

To add polynomials, combine all like terms.

Adding Polynomials

Add: (3x – 8) + (4x2 – 3x + 3).

= 4x2 + 3x – 3x – 8 + 3

= 4x2 – 5

= 3x – 8 + 4x2 – 3x + 3

Example

(3x – 8) + (4x2 – 3x + 3)

Add: using a vertical format.

Example

3 2 28 4 5 and 5 1y y y

3 2

2

3 2

8 4 5

5 1

8 6

y y

y

y y

To Subtract Polynomials

To subtract a polynomial, add its opposite.

Subtracting Polynomials

Example

Subtract 4 – (– y – 4).

4 – (– y – 4) = 4 + y + 4

= y + 4 + 4

= y + 8

Example

= 3a2 – 6a + 11

(– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)

= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7

= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7

Subtract (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7).

Subtract:

Example

3 2 3 2

3 2 3 2

3 2 3 2

3 2

(2 8 7 ) (3 2 3)

(2 8 7 ) ( 3 2 3)

2 8 7 3 2 3

10 7 3

x x x x x

x x x x x

x x x x x

x x x

3 2 3 2(2 8 7 ) (3 2 3)x x x x x

Subtract:

Example

2 2 2 2 2

2 2 2 2

9 7 4 6 3 4 10

9 4 4 10 4

a b ab ab b a ab b

a b ab ab b

2 2 2 2 2(9 7 4 ) (6 3 4 10 )a b ab ab b a ab b

Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function.

A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses.

A polynomial function of degree 2 is a quadratic function.

In general, for the quadratic equation of the form y = ax2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0.

a > 0x

a < 0x

Types of Polynomials

Polynomial functions of degree 3 are cubic functions. Cubic functions have four different forms, depending on the coefficient of the x3 term.

x3 coefficient is positive

x3 coefficient is negative

Types of Polynomials