Book cover for Intermediate Algebra

Intermediate Algebra

Julie Miller, Molly O'Neill, Nancy Hyde

ISBN #9781259610233

5th Edition

5,614 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on Polynomials 4 encapsulates advanced topics in polynomial functions including division techniques such as synthetic division, and essential theorems like the Remainder and Factor Theorems. Emphasis is placed on factoring techniques, polynomial evaluation, and analysis of graphs, which are essential for solving polynomial equations and understanding complex functions in both academic problems and real-world applications.

Learning Objectives

1

Understand the structure and properties of polynomials, including degree, leading coefficient, and standard form.

2

Learn and apply techniques for polynomial division, including synthetic division and long division.

3

Utilize the Remainder Theorem and Factor Theorem to evaluate and factor polynomials.

4

Analyze and interpret the behavior of polynomial functions through their graphs.

5

Solve complex polynomial equations and apply these methods to real-world problems.

Key Concepts

CONCEPT

DEFINITION

Polynomial

An expression consisting of variables and coefficients, composed using only addition, subtraction, and multiplication, and where variables have non-negative integer exponents.

Degree

The highest exponent of the variable in a polynomial, which dictates the polynomial’s overall behavior and number of possible roots.

Leading Coefficient

The coefficient of the term with the highest degree in a polynomial, important for determining the end behavior of the function.

Remainder Theorem

A theorem stating that the remainder of the division of a polynomial P(x) by (x - r) is equal to P(r).

Factor Theorem

A special case of the Remainder Theorem which indicates that (x - r) is a factor of the polynomial P(x) if and only if P(r) = 0.

Synthetic Division

A streamlined process for dividing a polynomial by a binomial of the form (x - r), used to simplify computations.

Fundamental Theorem of Algebra

A principle stating that every non-zero, single-variable polynomial with complex coefficients has exactly as many complex roots (counting multiplicities) as its degree.

Example Problems

Example 1

a. $A(n)$ ______ is used to show repeated multiplication of the base. b. For $b \neq 0,$ the expression $b^{0}$ is defined to be _______. c. For $b \neq 0,$ the expression $b^{-n}$ is defined as _________. d. A number expressed in the form $a \times 10^{n}$, where $1 \leq|a|<10$ and $n$ is an integer is said to be written in ___________ __________.

Example 2

Write the expressions in expanded form and simplify. $$ b^{4} \cdot b^{3} \text { and }\left(b^{4}\right)^{3} $$

Example 3

Write the expressions in expanded form and simplify. $$ a b^{3} \text { and }(a b)^{3} $$

Example 4

Write an example of each property. (Answers may vary.) $$b^{n} \cdot b^{m}=b^{n+m}$$

Example 5

Write an example of each property. (Answers may vary.) $$(a b)^{n}=a^{n} b^{n}$$
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Step-by-Step Explanations

QUESTION

Divide the polynomial P(x) = 2x³ - 3x² - 8x + 12 by (x - 2) using synthetic division.

STEP-BY-STEP ANSWER:

Step 1: Write the coefficients of P(x): 2, -3, -8, and 12.
Step 2: Write the root from the divisor (x - 2), which is 2, to the left side.
Step 3: Bring down the first coefficient, 2.
Step 4: Multiply 2 (the value brought down) by the root 2 to get 4. Write this beneath the second coefficient (-3).
Step 5: Add -3 and 4 to get 1. Write 1 as the new second coefficient.
Step 6: Multiply 1 by 2 to get 2. Write 2 beneath the third coefficient (-8).
Step 7: Add -8 and 2 to get -6. Write -6 as the third coefficient.
Step 8: Multiply -6 by 2 to get -12. Write -12 beneath the fourth coefficient (12).
Step 9: Add 12 and -12 to get 0, which is the remainder.
Final Answer: The quotient is 2x² + 1x - 6 and the remainder is 0, so (x - 2) is a factor of P(x).

Synthetic Division

QUESTION

Evaluate P(x) = 3x³ + x² - 5x + 7 at x = -1 using the Remainder Theorem.

STEP-BY-STEP ANSWER:

Step 1: Identify x = -1 as the value to substitute.
Step 2: Substitute -1 into the polynomial: 3(-1)³ + (-1)² - 5(-1) + 7.
Step 3: Compute each term: 3(-1) = -3, (-1)² = 1, -5(-1) = 5, then add 7.
Step 4: Sum the results: -3 + 1 + 5 + 7 = 10.
Final Answer: P(-1) = 10, so the remainder when divided by (x + 1) would be 10.

Using the Remainder Theorem

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Common Mistakes

  • Failing to align polynomial terms in descending order before performing division.
  • Incorrectly handling negative signs during synthetic division additions and multiplications.
  • Confusing the Remainder Theorem's result with the quotient rather than seeing it as the remainder.
  • Overlooking the necessity to check for all possible factors when fully factoring a polynomial.