Review the new concepts in each section of the chapter and try to relate the mathematical meaning to each of the given terms.
Term
Mathematical Meaning
A. multiplicity
V
B. synthetic division
I
C. Conjugate Root Theorem
II
D. relative maximum
IV
E. polynomial
III
Practice makes perfect
We are given various mathematical terms.
A.multiplicity B.synthetic division [0.5em]
C.Conjugate Root Theorem [0.5em]
D.relative maximum E.polynomial
We will use the given mathematical terms and decide which meaning corresponds to each of them.
Mathematical Term A
We can write any polynomial function f(x) = x^n+a_(n-1)x^(n-1)+...+a_1x+a_0 in its factored form.
f(x) = (x-x_1)(x-x_2)...(x-x_n)
In this form, x_1, x_2, ..., x_n can be identified as the zeros of the function. If the binomial factor (x-x_i) appears 2 times, we say that the zero x_i has a multiplicity of 2. If it appears 3 times, then it has a multiplicity of 3, and so on.
The multiplicity is the number of times a zero is repeated in a polynomial function.
Synthetic division is the process of dividing a polynomial by a linear factor, omitting all variables and exponents.
Therefore, term B matches with Meaning I.
Mathematical Term C
Assume that we have a polynomial function with rational coefficients. If this function has a complex root, the Conjugate Root Theorem tells us that this root must come in a conjugate pair.
Conjugate Root Theorem
If a+bi is an irrational root where a and b are real numbers then a-bi is also a root.
Therefore, term C matches with Meaning II.
Mathematical Term D
If a function has a turning point where it changes its behavior from increasing to decreasing, we say that the y-coordinate of this point is a relative maximum of the function. Let's consider the following graph.
Notice that (- 2,3) is the highest point on the graph for all x-values less than 1. Therefore, (- 2,3) is a relative maximum.
A relative maximum is the greatest y-value in a region of a graph.
