5.2 Polynomials, Linear Factors, and Zeros

5.2 POLYNOMIALS, LINEAR FACTORS, AND ZEROS

GOALS FOR CLASS:

• ANALYZE THE FACTORED FORM OF A POLYNOMIAL

• WRITE A POLYNOMIAL FUNCTION FROM ITS ZEROS

• WHAT EFFECT MULTIPLE ZEROS HAVE ON A GRAPH

• FINDING A RELATIVE MIN AND MAX

I. WRITING A POLYNOMIAL IN FACTORED FORM

𝑥3−2𝑥2−15𝑥1.Remember: Steps for FactoringGCF is ALWAYS first

Then, look for Special PatternsFactors of c that sum to b, if a = 1

orFactors of ac that sum to b

𝑥 (𝑥2−2 𝑥−15)𝑥 (𝑥−5)(𝑥+3)

2. 3 𝑥 (4 𝑥2+4 𝑥+1)

3 𝑥 (2𝑥+1)23 𝑥 (2𝑥+1)(2 𝑥+1)

If we had to solve these two equations, what would the solutions AKA x-intercepts AKA zeros AKA roots AKA and answers be?

x = 0, x = 5, x = -3

x = 0, x = -1/2

II. FINDING ZEROS OF A POLYNOMIAL, AND SKETCHING THE GRAPH

3.

𝑥=−1 , 𝑥=2 ,𝑥=−3These are the zeros of our polynomialNow we need to find the

end behavior between each intervalWe can pick any point between our

zeros to see if the graph is above or below the x-axis. We also need to find out what our graph does as x goes to the left and right.X-value -4 -2 0 3

Y-value

TRY: FACTOR THE POLYNOMIAL AND FIND THE ZEROS, PLOT THE ZEROS, THEN SKETCH THE GRAPH.ALL WITHOUT A CALCULATOR!!!

4.

HOMEWORK

• PAGE 682#7-25 ODD

III. WRITING A POLYNOMIAL FROM ITS ZEROS

5. 6. 𝑥3−8 𝑥2+19 𝑥−12 𝑥4−9 𝑥3+27 𝑥2−31 𝑥+12

Now what is the difference in there graphs if they each only intersect the x-axis three times???

IV. MULTIPLE ZEROS, MULTIPLICITY, AND HOW MULTIPLE ZEROS AFFECT THE GRAPH

In example 6, we had multiple zeros, since x = 1 showed up twice. Therefore, 1 has a multiplicity of 2, since it shows up two times. If a number shows up as a solution three times, then it would have multiplicity of 3.

The multiplicity of a zero effects the shape of the graph.Example 5 had three zeros that all had multiplicity of 1, meaning that the graph looks linear through those there points.

Example 6 had two zeros with multiplicity of 1 (3 and 4) and 1 with multiplicity of 2 (x = 1). The graph looks quadratic through at x =1.

TRY: FIND THE ZEROS, STATE THE MULTIPLICITY OF EACH, THEN SKETCH THE GRAPH

7.

𝑥=0 ,𝑥=2 , 𝑥=2

X = 0 has multiplicity of 1 which means it will be linear

X = 2 has multiplicity of 2 which means it will be quadratic

CHALLENGE:8. GRAPH

V. IDENTIFYING A RELATIVE MINIMUM AND MAXIMUM

If the graph of a polynomial function has several turning points, then it may have a relative minimum or a relative maximum.

FIND THE RELATIVE MIN AND MAX OF THE FOLLOWING:

9. 10.

VI. USING A POLYNOMIAL FUNCTION IN REAL LIFE2 MINUTES TO SOLVE

VI. USING A POLYNOMIAL FUNCTION IN REAL LIFEWITH A PARTNER – 4 MINUTES - CHALLENGE

WITH A PARTNER: PAGE #1-6 (5 MINUTES TO COMPLETE, STARTING NOW)

HOMEWORK

• PG 683 #27-39 ODD, 40-42, 44, 47-54