Q6 (402onsider the polynomial f(x) = (x-a)³(x - b)²(x - c), for real numbers a, b, and c. Read each statement below and determine if it is true or false. f(x) is a polynomial of degree 5. f(x) has only real zeros. f(x) passes through the x-axis at x = c, but bounces off the x-axis at x = b and x = a. • f(x) has 5 turning points. ● As x→∞o, f(x) → ∞ As x-∞, f(x) →→∞ f(x) has a y-intercept at y = -(a*b* c) • f(x) has a positive leading coefficient ● f(x) has only positive zeros. Count how many statements you marked as TRUE: ● ● ● ● ●
Q6 (402onsider the polynomial f(x) = (x-a)³(x - b)²(x - c), for real numbers a, b, and c. Read each statement below and determine if it is true or false. f(x) is a polynomial of degree 5. f(x) has only real zeros. f(x) passes through the x-axis at x = c, but bounces off the x-axis at x = b and x = a. • f(x) has 5 turning points. ● As x→∞o, f(x) → ∞ As x-∞, f(x) →→∞ f(x) has a y-intercept at y = -(a*b* c) • f(x) has a positive leading coefficient ● f(x) has only positive zeros. Count how many statements you marked as TRUE: ● ● ● ● ●
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Polynomial Analysis and Multiple Choice Question
#### Question 6 (400)
Consider the polynomial \( f(x) = (x-a)^3(x-b)^2(x-c) \), for real numbers \( a, b, \) and \( c \).
Read each statement below and determine if it is true or false.
- \( f(x) \) is a polynomial of degree 5.
- \( f(x) \) has only real zeros.
- \( f(x) \) passes through the x-axis at \( x = c \), but bounces off the x-axis at \( x = b \) and \( x = a \).
- \( f(x) \) has 5 turning points.
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- \( f(x) \) has a y-intercept at \( y = -(a \times b \times c) \).
- \( f(x) \) has a positive leading coefficient.
- \( f(x) \) has only positive zeros.
Count how many statements you marked as TRUE: ____________
#### Question 7 (200)
Multiple Choice: Let \( h(x) \) be a polynomial with 3 real zeros and two imaginary zeros. Which of the following could be the equation of \( h(x) \)?
A) \( h(x) = (x-2)(x-5)(x+7)(x^2 - 7x + 12) \)
B) \( h(x) = (x-3)^3(x^2 - 2x + 2) \)
C) \( h(x) = x^6 - 5x^3 + 3x^2 \)
D) \( h(x) = (x^2 - 9)(x^2 - (2 + 3i)) \)
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This content helps in understanding polynomial functions, analyzing graphical behavior, and recognizing zeroes—both real and imaginary.
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