Algebra 2 by Richard Wright
Are you not my student and
has this helped you?
Objectives:
SDA NAD Content Standards (2018): AII.4.3, AII.5.1, AII.5.3, AII.6.3
A landscape company is going to put some decorative rectangular prism-shaped stepping stones to make a path across a creek. Each stone will use 648 cubic inches of cement because that is convenient based on their cement supply. They decided that having the width be six inches greater than the length is a pleasing proportion and that the height should be one fourth the width for strength. What should be the dimensions of the stepping stone?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the stepping stone. This lesson highlights a variety of tools for writing polynomial functions and solving polynomial equations.
The Remainder Theorem provides a convenient way to evaluate polynomials based on division. A polynomial may be evaluated at f(k) by dividing it by x − k. In other words, f(k) is the remainder obtained by dividing f(x) by x − k. Synthetic division makes the process quick.
If a polynomial f(x) is divided by x − k, then the remainder is the value f(k).
To evaluate polynomial f(x) at x = k using the Remainder Theorem,
Use the Remainder Theorem to evaluate f(x) = x4 − 3x3 − x2 + 2x − 13 at x = 2.
Solution
To use the Remainder Theorem, use synthetic division to divide the polynomial by x − 2.
$$ \begin{array}{rrrrrr} \underline{2}| & 1 & -3 & -1 & 2 & -13 \\ & & 2 & -2 & -6 & -8\\ \hline & 1 & -1 & -3 & -4 & |\underline{-21} \end{array} $$
The remainder is −21. Therefore, f(2) = −21.
Analysis
It is possible to check the answer by evaluating f(2).
f(x) = x4 − 3x3 − x2 + 2x − 13
f(2) = (2)4 − 3(2)3 − (2)2 + 2(2) − 13
= −21
Use the Remainder Theorem to evaluate f(x) = 3x5 − x4 − 2x3 + x2 + 3 at x = 1.
Solution
f(1) = 4
The Factor Theorem says that if (x − k) is a factor of a function, then x = k is a zero of the function. A zero is a value of x that makes f(x) = 0.
It turn out that a polynomial of degree n in the complex number system will have n zeros. The Factor Theorem can be used to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, its zeros can easily be found.
According to the Factor Theorem, k is a zero of f(x) if and only if (x − k) is a factor of f(x).
To solve a polynomial equation given one factor using the factor theorem,
Show that (x − 1) is a factor of x3 − 2x2 − 5x + 6. Find the remaining factors. Use the factors to determine the zeros of the polynomial.
Solution
Use synthetic division to show that (x − 1) is a factor of the polynomial.
$$ \begin{array}{rrrrr} \underline{1}| & 1 & -2 & -5 & 6 \\ & & 1 & -1 & -6 \\ \hline & 1 & -1 & -6 & |\underline{\phantom{0}0} \end{array} $$
The remainder is zero, so (x − 1) is a factor of the polynomial. The quotient is x2 − x − 6 which is a quadratic. Factor that quadratic.
x2 − x − 6 = (x + 2)(x − 3)
Set each factor, including the given one, equal to zero and solve for x.
x − 1 = 0
x = 1
Or
x + 2 = 0
x = −2
Or
x − 3 = 0
x = 3
The zeros of x3 − 2x2 − 5x + 6 are −2, 1, and 3.
Use the Factor Theorem to find the zeros of f(x) = x3 − 5x2 − 10x + 24 given that (x − 4) is a factor of the polynomial.
The zeros are −2, 3, and 4.
Use the remainder theorem to evaluate f(x) at the given x value.
Show that the given binomial is a factor of f(x), then find the zeros of f(x).
Mixed Review
Solve by factoring.