Finding Zeros of Polynomials
Find the Zeros:
x2− 4x+ 3= 0
x= 1, 3
What happens 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
To find the Horizontal Asymptote of a polynomial, what do you have to look at?
The degree of the numerator and denominator
*Equal Degree= Coefficients, BOBO, BOTNO*
How do you build the LCD (least common denominator)?
Factor out all of the denominators and bring in any factors you don't already have on the LCD line
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)
How do you find the Vertical Asymptote of a polynomial?
Sent the denominator equal to zero and solve!
Solve:
(x-3) / (x+1) = (4x-6) / (x+1)(x+2)
x= 0, 5
What does the degree of a polynomial tell about the number of zeros found on the graph?
The degree of a polynomial tells 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
Can you have a Slant Asymptote if you also have a Horizontal Asymptote?
No, in order to have a Slant Asymptote you cannot have a Horizontal Asymptote
Solve:
(x) / (x+3) = (3) / (x+2)(x+3)
x= 1
x= -3 is extraneous
Find the Polynomial:
x= -4, -2, 5
y-intercept of (0, 8)
y= -1/5 (x-5) (x+2) (x+4)
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)
Are (x+2) and (x+1) solutions to x3- 2x2- 5x+ 6?
*use synthetic division
(x+2) is a solution (no remainder), (x+1) is not a solution (has a remainder)
How do you find the equation of the Slant Asymptote line?
Divide the denominator into the numerator and drop the remainder
(using synthetic division speeds this process up)
What makes a value of X qualify as an extraneous solution?
If a value of X causes a zero in the denominator, it is an extraneous solution
Determine the number of Real Zeros:
y= x5- 15x3+ 10x2+ 60x- 72
*hint: use your CALCULATOR
2 real zeros
x= -3, 2
Is The Lorax the best children's movie ever made?
YES
What is the best kind of cookie known to man?
Chocolate Chip
Find the hole in the equation:
(x2- x- 20) / (x2+ 5x+ 4)
Hole is at (-4, 3)
*plug -4 into simplified equation to find Y*
Solve:
- (x2+x-8) / (x2+2x-15) = (x) / (x-3)
x= -4, 1