5.2 Polynomials, Linear Factors, and Zeros

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Aim: How do we find the zeros of polynomial functions?

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Presentation transcript:

5.2 Polynomials, Linear Factors, and Zeros

Write a Polynomial in Factored Form What is the factored form of x3 – 2x2 – 15x

Roots, Zeros, & X-Intercepts

Find the Zeros of a Polynomial Function What are the zeros of y = (x + 2)(x – 1)(x – 3)? Graph the Function.

Factor Theorem 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.

Write a Polynomial Function from its Zeros What is the cubic polynomial function in standard form with zeros -2, 2, and 3?

Write a Polynomial Function from its Zeros What is the quartic polynomial function in standard form with zeros -2, -2, 2, and 3?

Graph both functions Graph the two functions from the previous slides. How do they differ? How are they similar?

x-intercept a will be close to linear if Multiple Zeros If a is a zero of multiplicity n in the polynomial function y = P(x), then the behavior of the graph at the x-intercept a will be close to linear if n = 1, close to quadratic if n = 2, close to cubic if n = 3, and so on.

A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs.

Example What are the zeros of f(x) = x4 – 2x3 – 8x2? What are their multiplicities? How does the graph behave at these zeros?

Relative Extrema If the graph of a polynomial function has several turning points, the function can have a relative maximum and a relative minimum.

Example What are the relative maximum and minimum of f(x) = x3 + 3x2 – 24x

Using a Polynomial Function to Maximize Volume The design of a digital box camera maximizes the volume while keeping the sum of the dimensions at 6 inches. If the length must be 1.5 times the height, what should each dimension be?

Exit Ticket What is the factored form of x3 – x2 – 12x? 2. What are the zeros of y = x(x – 3)(x + 5)? Graph the function. What is a cubic function with zeros 3, 3 and -3? What are the zeros of f(x) = x3 – 4x2 + 4x? What are the multiplicities? 5. What are the relative maximum and minimum of f(x) = 3x3 + x2 – 5x?