Factoring:
Writing Polynomials in Factored Form:
Example:
Zeroes:
To find the zeroes of a polynomial function, you must use the zero product property.
Zero-Product Property: if the product of two or more factors is zero, then one of the factors must be zero
Factor Theorem:
Factor Theorem: where the expression x - a is a factor of a polynomial if and only if the value a is a zero of the related polynomial function.
Writing a polynomial function from its Zeroes uses the Factor Theorem.
Example:
Multiplicities:
Multiplicity: number of times the related linear factor is repeated in the factored form of the polynomial
-to look at the number's multiplicity look at the degree it is raised to.
How to find a Multiplicity of a zero:
Relative Minimums/Maximums:
Relative Minimum: the value of the function at a down-to-up turning point, the lowest point in the graph
Relative Maximum: the value of the function at an up-to-down turning point, the highest point found in the graph
-you can use your graphing calculator to help find the maximum and minimum of the graph.
Maximizing the Volume: using polynomial functions
-maximizing the volume is perfect for those real-life situations that are needed to be solved
Here is an Example:
- The design of a digital box camera maximizes the volume while the sum of the dimensions at 4 inches. If the length must be 1.5 times the height, what should each dimension be?
Step 1. What is x? Step 2. What is the: -length -width Step 3. Create the Polynomial Step 4. When entering the polynomial into the graph use the maximum button found on the graphing calculator . We found that the height is 1.6 inches, which is x so substitute to find the rest of the dimensions. | Step 1. x=the height of the camera Step 2. length = 1.5x width = 6 - (x + 1.5x) = 6 - 2.5x Step 3. V = (length)(width)(height) V = (1.5x)(6 - 2.5x)(x) V = -3.75x3 + 9x2 Step 4.: As you can see the height is 1.6 inches, the length is 2.4 inches, and the width is 2 inches.
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Extra Practice
1. Write a polynomial function with the given zeroes: x=-2,1,4
2. Find the zeroes of the function. State the multiplicity of any multiple zeroes. y=x4-8x2+16
3. Use a graphing calculator to find the relative maximum, relative minimum, and zeroes of each function. y=5x3+x2-9x+4
