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Quiz 4 in Math 121: Polynomial Behavior, Roots, and Factors, Quizzes of Algebra

The instructions and problems for quiz 4 in math 121, focusing on determining the behavior of polynomials at extreme values, finding x-intercepts, using the rational zero theorem and descartes' rule of signs to find possible zeros, and factoring a polynomial.

Typology: Quizzes

Pre 2010

Uploaded on 08/18/2009

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Math 121, Quiz 4, 25 October 2004
Name:
Instructions. Complete questions 1(a), 2, 3(a) and 4. The others are for practice.
1. Determine the far right and far left behavior of the polynomials:
(a) P(x) = x7+ 2000000x5+ 7x2
12.
(b) Q(x) = 10x4
50000x3+ 7.
2. Find the x-intercepts of P(x) = (x3000)3(x2000)2(x+ 1000)1(x+ 3000)2000. For each
intercept, determine whether the graph of P(x) crosses or merely touches the x-axis.
3. Consider the polynomial P(x) = 6x5
3x4+ 12x2+ 13x+ 4.
(a) Use the Rational Zero Theorem to list the possible rational zeros of P(x)
(b) Use the Zero Location Theorem to prove there is a zero between x=1 and x= 0. Do
not find that zero.
(c) Use Descartes’ Rule of signs to determine the number of positive and the number of negative
real zeros that P(x) may posses.
4. Using the information that the polynomial P(x) = 2x4
x3
3x2
31x15 has factors
(x+1
2) and (x3), find all zeros of P(x).

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Math 121, Quiz 4, 25 October 2004

Name:

Instructions. Complete questions 1(a), 2, 3(a) and 4. The others are for practice.

  1. Determine the far right and far left behavior of the polynomials: (a) P (x) = −x^7 + 2000000x^5 + 7x^2 − 12. (b) Q(x) = 10x^4 − 50000 x^3 + 7.
  2. Find the x-intercepts of P (x) = (x − 3000)^3 (x − 2000)^2 (x + 1000)^1 (x + 3000)^2000. For each intercept, determine whether the graph of P (x) crosses or merely touches the x-axis.
  3. Consider the polynomial P (x) = 6x^5 − 3 x^4 + 12x^2 + 13x + 4. (a) Use the Rational Zero Theorem to list the possible rational zeros of P (x) (b) Use the Zero Location Theorem to prove there is a zero between x = −1 and x = 0. Do not find that zero. (c) Use Descartes’ Rule of signs to determine the number of positive and the number of negative real zeros that P (x) may posses.
  4. Using the information that the polynomial P (x) = 2x^4 − x^3 − 3 x^2 − 31 x − 15 has factors (x + 12 ) and (x − 3), find all zeros of P (x).