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Explain why a polynomial of degree 3 has at least one root. Start by examining the end-behavior of a polynomial of degree 3. Which statement correctly describes the end-behavior of a polynomial of degree 3? OA If f(x) has a leading term of ax, then lim f(x) = lim f(x) = b for some real number b. X00 X - 00 OR If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = - o when a>0 or lim f(x) = - 00 and lim f(x) = 00 when a <0. X00 X- 00 X00 X - 00 Oc If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = 00 when a>0 or lim f(x) = - 00 and lim f(x) = - 00 when a <0. X- 00 X00 X - 00 Which statement correctly concludes that f has at least one root? O A. Since fis a continuous function and the limits at infinity are both finite values, the intermediate-value theorem guarantees that f(R) = (-b, b) for some real number b and, hence, f has at least one root. O B. Since fis a continuous function and the limits at infinity have opposite signs, the intermediate-value theorem guarantees that f(R) =R and, hence, f has at least one root. O C. Since fis a continuous function and the limits at infinity have the same sign, the intermediate-value theorem guarantees that f(R) = [0, 00) or f(R) = (- 00, 0] and, hence, f has at least one root.

Explain why a polynomial of degree 3 has at least one root. Start by examining the end-behavior of a polynomial of degree 3. Which statement correctly describes the end-behavior of a polynomial of degree 3? OA If f(x) has a leading term of ax, then lim f(x) = lim f(x) = b for some real number b. X00 X - 00 OR If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = - o when a>0 or lim f(x) = - 00 and lim f(x) = 00 when a <0. X00 X- 00 X00 X - 00 Oc If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = 00 when a>0 or lim f(x) = - 00 and lim f(x) = - 00 when a <0. X- 00 X00 X - 00 Which statement correctly concludes that f has at least one root? O A. Since fis a continuous function and the limits at infinity are both finite values, the intermediate-value theorem guarantees that f(R) = (-b, b) for some real number b and, hence, f has at least one root. O B. Since fis a continuous function and the limits at infinity have opposite signs, the intermediate-value theorem guarantees that f(R) =R and, hence, f has at least one root. O C. Since fis a continuous function and the limits at infinity have the same sign, the intermediate-value theorem guarantees that f(R) = [0, 00) or f(R) = (- 00, 0] and, hence, f has at least one root.

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.4: Graphing Polynomial Functions
Problem 47PS
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Explain why a polynomial of degree 3 has at least one root. Start by examining the end-behavior of a polynomial of degree 3. Which statement correctly describes the end-behavior of a polynomial of degree 3? OA If f(x) has a leading term of ax, then lim f(x) = lim f(x) = b for some real number b. X00 X - 00 OP If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = - 0 when a>0 or lim f(x) = - 0o and lim f(x) = oo when a <0. X00 X - 00 X00 X - 00 Oc If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = 00 when a>0 or lim f(x) = - 00 and lim f(x) = - oo when a <0. X00 X - 00 X00 X - 00 Which statement correctly concludes that f has at least one root? O A. Since fis a continuous function and the limits at infinity are both finite values, the intermediate-value theorem guarantees that f(R) = (- b, b) for some real number b and, hence, f has at least one root. O B. Since fis a continuous function and the limits at infinity have opposite signs, the intermediate-value theorem guarantees that f(R) = R and, hence, f has at least one root. O C. Since fis a continuous function and the limits at infinity have the same sign, the intermediate-value theorem guarantees that f(R) = [0, 00) or f(R) = (- 00, 0] and, hence, f has at least one root.
Transcribed Image Text:Explain why a polynomial of degree 3 has at least one root. Start by examining the end-behavior of a polynomial of degree 3. Which statement correctly describes the end-behavior of a polynomial of degree 3? OA If f(x) has a leading term of ax, then lim f(x) = lim f(x) = b for some real number b. X00 X - 00 OP If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = - 0 when a>0 or lim f(x) = - 0o and lim f(x) = oo when a <0. X00 X - 00 X00 X - 00 Oc If f(x) has a leading term of ax, then either lim f(x) = 00 and lim f(x) = 00 when a>0 or lim f(x) = - 00 and lim f(x) = - oo when a <0. X00 X - 00 X00 X - 00 Which statement correctly concludes that f has at least one root? O A. Since fis a continuous function and the limits at infinity are both finite values, the intermediate-value theorem guarantees that f(R) = (- b, b) for some real number b and, hence, f has at least one root. O B. Since fis a continuous function and the limits at infinity have opposite signs, the intermediate-value theorem guarantees that f(R) = R and, hence, f has at least one root. O C. Since fis a continuous function and the limits at infinity have the same sign, the intermediate-value theorem guarantees that f(R) = [0, 00) or f(R) = (- 00, 0] and, hence, f has at least one root.
Expert Solution
Step 1 : end behaviour

When a > 0limx→-∞ f(x) →-∞limx→∞ f(x)    →∞

Advanced Math homework question answer, step 1, image 1

When a < 0limx→-∞ f(x) →∞limx→∞f(x) →-∞

Advanced Math homework question answer, step 1, image 2

Hence option 'b' is correct.

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