MAC 1140 – Section 3.3 – Roots of Polynomial Equations
Notes:
The zeros of a polynomial function y = p(x) are the roots (solutions) of the polynomial equation
0 = p(x).
A polynomial equation of degree n has n roots.
The x-intercepts of a polynomial function y = p(x) are the REAL roots of the polynomial
equation 0 = p(x).
If a + bi is a root of a polynomial equation, so is a – bi
Find all the roots of the following polynomial equations, stating multiplicity (number of times root is a
solution) when multiplicity is greater than one. Also state the degree of the polynomial.
A. By Factoring:
1. x4 – 25 = 0
2. x3 – 2x2 - 3x = 0
3. x3 – 2x2 + 3x = 0
B. By Factoring, Roots of Multiplicity:
1. x2 – 6x + 9 = 0
2. 8x4 + 64x2 = 0
3. (x – 3)2(x + 1)3 = 0
Graph each equation in B on graphing calculator and sketch its graph below.
NOTE: A root of even multiplicity corresponds to a "touch and turn" on the graph.
A root of odd multiplicity (3 or higher) corresponds to an apparent "flat spot" on the
graph.
C. Solve giving exact answers (Use graphing calculator for observed root(s); then Synthetic
Division.)
1. 2x3 - 3x2 - 4x + 1 = 0
2. x4 - 5x3 + 2x2 + 7x - 5 = 0
3. x4 – x3 – 3x2 – 3x – 18 = 0
D. Determine all real solutions to the nearest hundredth. Remember: Real solutions or roots are
x-intercepts. Let Y1= left side of equation and Y2 = right side of equation. Use 2nd, Trace (for
Calc), 5 for intersection. You must be able to see a point of intersection to do this. If you do not
see a point where the graph intersects the x-axis, enlarge your window.
1. x4 – 3x2 + 6x – 24 = 0
2. x3 – 4x = -6
For the two problems below find all real roots to the nearest hundredth as we did in D above:
1. 2x3 – 5x2 – 6x + 4 = 0
2. 3x4 – 5x3 – x2 – 8x + 4 = 0
Answers to 1: -1.24, 0.5, 3.24
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