1. By: Deanna Carbone, Jacqueline DiSalvatore, and Alyssa Fanelli
Graphing Polynomials
By: Deanna Carbone, Jacqueline DiSalvatore, and Alyssa Fanelli

2. Definitions Real Zeros – the value of x when y=0; the y-intercepts
Rational Roots Theorem – p/q is a rational zero of f, a polynomial of degree 1 or higher or the coefficient is an integer
Multiplicity – when there are multiple of the same real zeros [ex: (x+2)2]
Sign Analysis – testing the intervals for values between the real zeros
Maximum – highest y-value and turning point of the graph
Minimum – lowest y-value and turning point of the graph
Y-intercept – point on the graph when y=0
Intervals of increasing and decreasing – where y increases or decreases on the graph; determined by the change in y, but written using x-values
Degree of a polynomial – highest exponent in the equation

3. Step 1 Find the real zeros
Factor the equation completely (use Rational Roots Theorem if needed)
Look for multiplicities
Even multiplicity – graph touches the x-axis
Odd multiplicity – graph goes through the x-axis

4. Step 2
Test intervals
Choose values between the real zeros found from Step 1
Plug these values into each factor of the original polynomial and determine whether the answer is positive or negative
Once all values are tested, multiply the signs of each answer to determine the end behavior
The end behavior should relate to the degree of the polynomial
Even degree – same end behavior
Odd degree – opposite end behavior

5. Step 3 Determine the number of turning points
Subtract 1 from the degree of the polynomial to determine the number of maximums and minimums

6. Step 4 Graph the polynomial
Find the y-intercept by plugging 0 into the original equation
Graph
Analyze the graph
X-intercepts
Maximums and minimums (from calculator)
Intervals of increasing and decreasing

7. Odd Degree P(x) = x3+8x2+11x-20 = (x+4)(x+5)(x-1) Step 1
Find real zeros: x= -4, x= -5, x=1
No multiplicities

8. Odd Degree (cont’d) Step 2 Test intervals -6 -4.5 0 2 x+4 - - + + x x x

9. Odd Degree (cont’d) Step 3 Step 4 Find the turning points (Degree – 1)
Graph (on next slide)
Y-intercept = -20
X-intercepts = -4, -5, 1
Max: (-4.5, 1.4)
Min: (-.81, -24.2)
Intervals of increasing: (-∞, -4.5]
[-.81, ∞)
Interval of decreasing: [-4.5, -.81]

10. Odd Degree (cont’d)

11. Even Degree P(x) = x4-2x3-3x2 = x2(x2-2x-3) Step 1
Find real zeros: x=0, x=3, x= -1
0 has a multiplicity of 2

12. Even Degree (cont’d)
Step 2 Test intervals -2 -1/2 1 4 x x x

13. Even Degree (cont’d) Step 3 Step 4 Find the turning points (Degree-1)
Graph (on next slide)
Y-intercept = 0
X-intercepts = 0, 3, -1
Max: (0,0)
Mins: (-.68, -.544)
(2.18, -12.4)
Intervals of increasing: [-.686, 0]
[2.18, ∞)
Intervals of decreasing: (-∞, -.686]
[0, 2.18]

14. Even Degree (cont’d)

15. Even Degree with a Multiplicity
P(x) = x4+x3-12x2
= x2(x+4)(x-3)
Step 1
Find real zeros: x=0, x= -4, x=3
0 has a multiplicity of 2

16. Even Degree with a Multiplicity (cont’d)
Step 2
Test intervals x x x

17. Even Degree with a Multiplicity (cont’d)
Step 3
Find the turning points (Degree-1)
(4-1) = 3 turning points
Step 4
Graph (on next slide)
Y-intercept = 0
X-intercepts = -4, 3, 0
Max: (0,0)
Mins: (2.10, )
(-2.85, )
Intervals of increasing: [-2.85, 0], [2.10, ∞)
Intervals of decreasing: (-∞, -2.85], [0, 2.10]

18. Even Degree with a Multiplicity (cont’d)