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UNIT FOUR
UNIVERSITY OF THE PEOPLE
1201-01: COLLEGE ALGEBRA
Task 1: Polynomial Graph Interpretation
(i) Turning points, zeros, and x-intercepts:
The turning points are the local maximums and minimums: A approx (-1.569, -3.124), B approx (0.319, 8.643), and
C (2, 0).
, 2
The zeros and x-intercepts are where the graph hits the horizontal axis: D = (-2, 0), E = (-1, 0), and C = (2, 0).
(ii) Multiplicity:
Yes, the zero at x = 2 (Point C) has a multiplicity of 2.
Reason: The graph touches the x-axis and turns around (bounces) rather than crossing through it. This behaviour
indicates an even multiplicity.
(iii) Degree, Polynomial, and Intervals:
Degree: Since there are 3 distinct zeros and one has multiplicity 2, the minimum degree is 4.
Polynomial: f(x) = (x + 2)(x + 1)(x - 2)^2.
Increasing intervals: (-1.569, 0.319) and (2, infinity).
Decreasing intervals: (-infinity, -1.569) and (0.319, 2).
(iv) Local Maximum/Minimum:
Local Maximum: Point B (0.319, 8.643).
Local Minimums: Point A (-1.569, -3.124) and Point C (2, 0).
(v) Remainder when divided by (x - 4):
Using the Remainder Theorem, calculate f(4).
f(4) = (4 + 2)(4 + 1)(4 - 2)^2 = 6 * 5 * 4 = 120.
Task 2: f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7
(i) Zeros using Rational Root Theorem and Synthetic Division:
Possible roots are divisors of 7: +/- 1, +/- 7.
Testing x = 1: 1 - 8 - 8 + 8 + 7 = 0. So 1 is a zero.
Testing x = -1: 1 + 8 - 8 - 8 + 7 = 0. So -1 is a zero.
The remaining quadratic x^2 - 8x - 7 = 0 gives roots 4 + sqrt(23) and 4 - sqrt(23).
Zeros: 1, -1, 4 + sqrt(23), 4 - sqrt(23).
(ii) Graph:
UNIT FOUR
UNIVERSITY OF THE PEOPLE
1201-01: COLLEGE ALGEBRA
Task 1: Polynomial Graph Interpretation
(i) Turning points, zeros, and x-intercepts:
The turning points are the local maximums and minimums: A approx (-1.569, -3.124), B approx (0.319, 8.643), and
C (2, 0).
, 2
The zeros and x-intercepts are where the graph hits the horizontal axis: D = (-2, 0), E = (-1, 0), and C = (2, 0).
(ii) Multiplicity:
Yes, the zero at x = 2 (Point C) has a multiplicity of 2.
Reason: The graph touches the x-axis and turns around (bounces) rather than crossing through it. This behaviour
indicates an even multiplicity.
(iii) Degree, Polynomial, and Intervals:
Degree: Since there are 3 distinct zeros and one has multiplicity 2, the minimum degree is 4.
Polynomial: f(x) = (x + 2)(x + 1)(x - 2)^2.
Increasing intervals: (-1.569, 0.319) and (2, infinity).
Decreasing intervals: (-infinity, -1.569) and (0.319, 2).
(iv) Local Maximum/Minimum:
Local Maximum: Point B (0.319, 8.643).
Local Minimums: Point A (-1.569, -3.124) and Point C (2, 0).
(v) Remainder when divided by (x - 4):
Using the Remainder Theorem, calculate f(4).
f(4) = (4 + 2)(4 + 1)(4 - 2)^2 = 6 * 5 * 4 = 120.
Task 2: f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7
(i) Zeros using Rational Root Theorem and Synthetic Division:
Possible roots are divisors of 7: +/- 1, +/- 7.
Testing x = 1: 1 - 8 - 8 + 8 + 7 = 0. So 1 is a zero.
Testing x = -1: 1 + 8 - 8 - 8 + 7 = 0. So -1 is a zero.
The remaining quadratic x^2 - 8x - 7 = 0 gives roots 4 + sqrt(23) and 4 - sqrt(23).
Zeros: 1, -1, 4 + sqrt(23), 4 - sqrt(23).
(ii) Graph:
