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