section 5.2 – polynomials, linear factors, and zeros why??????????? a storage company needs to...
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Section 5.2 – Polynomials, Linear Factors, and Zeros
WHY???????????
A storage company needs to design a new storage box that has twice the volume of its
largest box. Its largest box is 5 feet long, 4 feet wide, and 3 feet high. The new box must be formed by increasing each dimension by the
same amount. Find the increase in each dimension.
Section 5.2 – Polynomials, Linear Factors, and Zeros
Students will be able to:
• Analyze the factored form of a polynomial
• Write a polynomial function from its zeros
Lesson Vocabulary
Factor Theorem Multiple Zero
Multiplicity Relative Minimum
Relative Maximum
Section 5.2 – Polynomials, Linear Factors, and Zeros
If P(x) is a polynomial function, the solutions of the related polynomial equation P(x) = 0 are the
zeros of the function
Essential Understanding:
Finding the zeros of a polynomial function will help you factor the polynomial, graph the function, and solve the related polynomial
equation.
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 1:
What is the factored form of x3 – 2x2 – 15x?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 1b:
What is the factored form of x3 – x2 – 12x?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 2:
What are the zeros of y = (x+2)(x-1)(x-3)?
Graph the function.
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 2b:
What are the zeros of y = x(x + 5)(x-3)?
Graph the function.
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 3:
What is a cubic polynomial function in standard form with zeros -2, 2, 3?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 3b:
What is a quartic polynomial function in standard form with zeros -2, -2, 2, and 3?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 3c:
Graph the two previous equations. How do the differ? How are they similar?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 4:
What is a quadratic polynomial function in standard form with zeros 3 and -3?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 4b:
What is a cubic polynomial function in standard form with zeros 3, 3, and -3?
Section 5.2 – Polynomials, Linear Factors, and Zeros
You can write the polynomial function in Problem 3 in factored form as
f(x) = (x + 2)(x – 2)(x – 3)and
g(x) = (x+2)2(x – 2)(x – 3).
In g(x) the repeated linear factor x + 2 makes -2 a multiple zero.
Since the linear factor x + 2 appears twice, you can say that -2 is a zero of multiplicity 2.
Section 5.2 – Polynomials, Linear Factors, and Zeros
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 5:
What are the zeros of f(x) = x4 – 2x3 – 8x2?
What are their multiplicities?
How does the graph behave at those zeros?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 5b:
What are the zeros of f(x) = x3 – 4x2 + 4x?
What are their multiplicities?
How does the graph behave at those zeros?
Section 5.2 – Polynomials, Linear Factors, and Zeros
If the graph of a polynomial function has several turning points, the function can have a relative minimum
and a relative maximum.
A relative max is the value of the function at an up-to-down turning point.
A relative min is the value of the function at a down-to-up turning point.
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 6:
What are the relative maximum and relative minimum of
f(x) = x3 +3x2 – 24x?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 6b:
What are the relative maximum and relative minimum of
f(x) = 3x3 +x2 – 5x?
Section 5.2 – Polynomials, Linear Factors, and Zeros
Problem 7:
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 that dimension be?