Finding Zeros of Polynomials
Find the Zeros:
x2− 4x+ 3= 0
x= 1, 3
What happens to the graph when a zero (x-value) has a multiplicity of 1?
The line crosses straight through the point on the graph.
Solve:
(2x3− x2− 25x− 12) / (x+3)
2x2− 7x− 4
What does the "a" value of an equation tell us about its graph?
The "a" value tells us whether a graph is positive or negative.
Find the Zeros:
2x2−2x−180=0
x= -9, 10
Find the Zeros and the Multiplicity:
6x2+ 36x+ 54= 0
x= -3
Multiplicity of 2
Solve:
(5x3− 22x2− 17x+ 11) / (x-5)
5x2+ 3x− 2 + (1) / (x-5)
Describe the End Behavior of this equation:
x4 - 2x2 + 1
Left side rises and right side rises
What does the degree of a polynomial indicate about the number of zeros found on the graph?
The degree of a polynomial indicates the maximum number of zeros
Find the Zeros and the Multiplicity:
y= x3- 2x2+ x
x= 0, 1
0 has a multiplicity of 1
1 has a multiplicity of 2
How do you know if (x-#) is a factor of a polynomial?
(x-#) is only a factor if:
1.) there is no remainder when divided out
2.) the output is zero when the value is plugged into the function
True or False:
If a Polynomial has an ODD degree, the End Behavior will be the SAME on both the left side of the graph and the right side.
FALSE
Polynomials with EVEN degrees will have an End Behavior that points in the SAME direction on both the left side of the graph and the right side
Find the Polynomial:
x= -3, 0, 3
The graph hits the point (2,5)
y= -1/2(x+3)(x)(x-3)
What happens to the graph of a polynomial when a zero has a multiplicity of 3?
The line on the graph crosses through the point and then lays flat (sometimes called a slide)
Solve:
(4x3 + 6x2 - 10x + 4) / (2x-1)
2x2 + 4x - 3 + (1) / (2x-1)
Describe the End Behavior of this Polynomial:
x3 − 4x2 + x + 6
Falls to the left and rises to the right
Determine the number of Real Zeros:
y= x5- 15x3+ 10x2+ 60x- 72
*hint: use your CALCULATOR
2 real zeros
x= -3, 2
Graph this Polynomial:
f(x)= (x+3)2 (x-2)2 (x+1)
We will work this one on the board :)
Solve Using Synthetic Division
(3x3 + 5x2 - 11x + 3) / (x+3)
3x2 - 4x + 1
Graph this Polynomial and describe the End Behavior:
f(x)= (2x-1)3 (x-2)2
We will graph this together
The End Behavior falls to the left and rises to the right
*there is a slide at x=1/2 and a bounce at x=2*