Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the fundamental operations with polynomials, including identifying types such as monomials, binomials, and trinomials, determining their degrees, and performing addition and subtraction by combining like terms. Additionally, it provides an introduction to exponent properties crucial for simplifying expressions, emphasizing that careful application of these properties underpins many real-world calculations, from evaluating functions to modeling complex systems like rocket trajectories.

Learning Objectives

1

Identify and classify polynomials, monomials, binomials, and trinomials.

2

Determine the degree of a polynomial and its individual terms.

3

Perform addition and subtraction of monomials and polynomials by combining like terms.

4

Evaluate polynomials for given variable values.

5

Simplify expressions using the multiplication properties of exponents and multiply monomials.

Key Concepts

CONCEPT

DEFINITION

Polynomial

An expression consisting of one or more terms in the form ax^m, combined using addition or subtraction.

Monomial

A polynomial with exactly one term, e.g., 3x² or -5a⁴.

Binomial

A polynomial with exactly two terms, e.g., 3a - 7 or y² - 9.

Trinomial

A polynomial with exactly three terms, e.g., x² - 5x + 6.

Degree of a Term

The exponent of the variable in that term; for a constant, it is 0.

Degree of a Polynomial

The highest degree among all terms in the polynomial.

Product Property of Exponents

For any base a, a^m * a^n = a^(m+n).

Power Property of Exponents

For any base a, (a^m)^n = a^(m*n).

Product to a Power Property

For any bases a and b, (ab)^n = a^n * b^n.

Example Problems

Example 1

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$5 x+2$$

Example 2

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$z^{2}-5 z-6$$

Example 3

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$a^{2}+9 a+18$$

Example 4

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$-12 p^{4}$$

Example 5

In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. $$y^{3}-8 y^{2}+2 y-16$$
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Step-by-Step Explanations

QUESTION

Find the degree of the polynomial: 3x³ - 5x + 7.

STEP-BY-STEP ANSWER:

Step 1: Identify the degree of each term. For 3x³, the degree is 3; for -5x, the degree is 1; for 7, the degree is 0.
Step 2: Determine the highest degree among these terms, which is 3.
Final Answer: The degree of the polynomial is 3.

Determine the Degree of a Polynomial

QUESTION

Simplify the addition: (4x² - 5x + 1) + (3x² - 8x - 9).

STEP-BY-STEP ANSWER:

Step 1: Remove parentheses and write all terms: 4x² - 5x + 1 + 3x² - 8x - 9.
Step 2: Group like terms: (4x² + 3x²) + (-5x - 8x) + (1 - 9).
Step 3: Combine coefficients: 7x² - 13x - 8.
Final Answer: 7x² - 13x - 8.

Adding Like Terms in Polynomials

QUESTION

Simplify the expression: (2^3) * (2^4).

STEP-BY-STEP ANSWER:

Step 1: Identify that the bases are the same (2) so apply the Product Property of Exponents.
Step 2: Add the exponents: 3 + 4 = 7.
Step 3: Write the expression with the new exponent: 2^7.
Final Answer: 2^7.

Simplify Using Exponent Properties

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Common Mistakes

  • Misclassifying polynomials by confusing the number of terms (e.g., calling a trinomial a polynomial without specifying its type when required).
  • Failing to correctly distribute negative signs when subtracting polynomials.
  • Combining unlike terms when adding or subtracting, which includes not only combining coefficients but leaving exponents unchanged.
  • Incorrectly applying exponent rules; for example, adding exponents when bases do not match or applying the power property incorrectly.