Chapter 4 Polynomials (visual)

62 terms · Algebra Structure and Method Book 1 / The Classic Richard G. Brown Mary P. Dolciani Robert H. Sorgenfrey William L. Cole

Polynomials

Exponents

Objective 4-1

To write and simplify expressions involving exponents.

a power of 5

First power of 5:

5 to the first power = 5

Second power of 5:

5 to the second power = 5 · 5

Third power of 5:

5 to the third power = 5 · 5 · 5

Fourth power of 5:

5 to the fourth power = 5 · 5 · 5 · 5

power of a number

The product
when a number is
multiplied by itself
a given number of times;
4 × 4 × 4, or 4³,
is the third power of 4.

base of a power

The number
that is used as a factor
a given number of times;
in 2⁵, 2 is the base.

exponent

In a power,
the number that indicates
how many times the base
is used as a factor;
in 6⁵, 5 is the exponent.

exponential form of a power

The expression n⁴ is
the exponential form of n ⋅n ⋅ n ⋅ n.

Caution: Be careful when an expression contains both

parentheses and exponents.

To simplify expressions that contain powers follow the steps used

to simplify numerical expressions.

Summary of Order of Operations

1. First simplify expressions within grouping symbols.

2. Then simplify powers.

3. Then simplify products and quotients
in order from left to right.

4. Then simplify sums and differences
in order from left to right.

monomial

An expression
that is either
a numeral, a variable,
or the product of a numeral and one or more variables.

constant (monomial)

A monomial consisting of a numeral only;
a term with no variable factor.

polynomial

A sum of monomials.

binomial

A polynomial of only two terms.

trinomial

A polynomial of only three terms.

coefficient

In the monomial 15a²b²,
15 is the coefficient or
numerical coefficient.

similar terms

Two monomials
that are exactly alike or
are the same
except for their numerical coefficients.
Also called like terms.

simplest form of a polynomial

A polynomial is in simplest form
when no two of its terms
are similar.

degree of a variable in a monomial

The number of times
that the variable occurs as a factor
in the monomial.

The sum of
the degrees of
the variables in
the monomial.

degree of a polynomial

The greatest of the degrees of its terms
after it has been simplified.

Multiplication

Lesson 4-3

Multiplying Monomials

Objective 4-3

To multiply monomials.

rule of exponents for products of powers

For all positive integers m and n,

a^m ⋅ a^n = a^(m+n)

To multiply two powers having the same base,

Lesson 4-4

Powers of Monomials

Objective 4-4

To find powers of monomials.

rule of exponents for a power of a power

For all positive integers m and n,

(a^m)^n = a^(m⋅n)

To find a power of a power,
you multiply the exponents.

rule of exponents for a power of a product

For every positive integer m,

(a⋅b)^m = a^m⋅b^m

To find a power of a product,
you find the power of each factor
and then multiply.

Lesson 4-5

Multiplying Polynomials by Monomials

Objective 4-5

To multiply a polynomial by a monomial.

Lesson 4-6

Multiplying Polynomials

Objective 4-6

To multiply polynomials.

Problem Solving

Lesson 4-7

Transforming Formulas

Objective 4-7

To transform a formula.

Lesson 4-8

Rate-Time-Distance Problems

Objective 4-8

To solve some word problems involving uniform motion.

uniform motion

Motion without change in speed or rate.

Area Problems

Objective 4-9

To solve some problems involving area.

Lesson 4-10

Problems Without Solutions

Objective 4-10

To recognize problems that do not have solutions.

Chapter 4 Summary

Polynomials Summary

Chapter 4 Summary 1.

The expression b
is an abbreviation for b⋅b⋅b⋅...⋅b.
(n factors)
The base is b and the exponent is n.

Chapter 4 Summary 2.

To simplify expressions
that contain powers
to simplify numerical expressions.

Summary of Order of Operations

1. First simplify expressions within grouping symbols.

2. Then simplify powers.

3. Then simplify products and quotients
in order from left to right.

4. Then simplify sums and differences
in order from left to right.

Chapter 4 Summary 3.

their similar terms.
Similar terms are monomials
that are exactly alike or
that differ only in
their numerical coefficients.

Chapter 4 Summary 4.

Rules of exponents:

a^m ⋅ a^n = a^(m+n)

(a^m)^n = a^(m⋅n)

(a⋅b)^m = a^m⋅b^m

Chapter 4 Summary 5.

Polynomials can be multiplied
in a vertical or horizontal form
by applying the distributive property:

ab + ac = a(b + c)
ba + ca = (b + c)a

ab − ac = a(b − c)
ba − ca = (b − c)a

Before multiplying,
it is wise
to rearrange the terms of
each polynomial
in order of
increasing or decreasing degree
in one variable.

Chapter 4 Summary 6.

A formula may be transformed
to express a particular variable
in terms of the other variables.

Chapter 4 Summary 7.

A chart can be used
to solve problems
Formulas to use are:
rate × time = distance
length × width = area of a rectangle

Chapter 4 Summary 8.

To solve problems involving area,
to make a sketch.

Chapter 4 Summary 9.

Problems may fail to have solutions
because of
lack of information,