College Algebra: Lesson 5.1 Introduction to Polynomial Equations and Graphs
1. A polynomial function can be written in the form p( x)  a n x n  a n 1 x n 1  ....  a1 x  a 0 , where n is a whole
number.
2. The largest power of a polynomial is called the degree of the polynomial.
3. A coefficient is the number in front of the variable, and the leading coefficient is the number in front of the
biggest power of x.
4. A solution of an equation is a value that makes the equation true when it is substituted into the equation.
5. To solve a quadratic equation, you can solve by factoring or use the quadratic formula. x 
 b  b 2  4ac
2a
The Leading Coefficient Test
You can determine the "end behavior" of a graph by looking at the degree and leading coefficient.
1. If the degree is even:
2. If the degree is odd:
a. positive leading coefficient
a. positive leading coefficient
b. negative leading coefficient
b. negative leading coefficient
Examples
Consider the following polynomial.
a. Determine the degree and the leading coefficient of r(x).
b. Describe the behavior of the graph of r(x) as x   .
r ( x)   ..as...x   and r ( x)   ..as...x  
1. r ( x)  8 x 2  8x  2
2. r ( x)  6 x 5  4 x 3  7 x 2  10  32
3. Does the given value of x solve the polynomial equation?
 59 x 2  3x  6 x 3  80; x  10
3. r ( x)  6 x 2 ( x  5)( x  1)
4. Solve the polynomial equation by factoring or using the quadratic formula, making sure to identify all the
solutions.
a. x 2  9 x  44  0
b. x 3  20 x  9 x 2
5. Solve the polynomial inequality. Enter your answer in interval notation.
a. ( x  4)( x  3)( 2  x)  0
b. x( x  5)( x  4)  0
c. x 2   x  20
d. x( x  5) 2 ( x  8)  0
e. 4 x 2  x  3
f. 3x 3  2 x 2  1
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6. Determine the x-intercepts, the y-intercept and the correct graph of the polynomial function. Enter all points
as ordered pairs, separate multiple answers with a comma, and select the graph from the options below.
a. s( x)  ( x  2)( x  1)(5  x)
b. f ( x)  x 2  x  2
c. r ( x)  ( x  2)( x  1)( x  1)( x  2)