Chapter 4 Polynomials (visual)
Chapter 4 Summary 9.
Problems may fail to have solutions because of lack of information, contradictory facts, or unrealistic results.
*exponent*
In a power, the number that indicates how many times the base is used as a factor; in 6⁵, 5 is the exponent.
*coefficient*
In the monomial 15a²b², 15 is the coefficient or *numerical coefficient.*
*uniform motion*
Motion without change in speed or rate.
To simplify expressions that contain powers follow the steps used
to simplify numerical expressions.
*constant (monomial)*
A monomial consisting of a numeral only; a term with no variable factor.
*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.*
Lesson *4-2*
*Adding and Subtracting Polynomials*
Chapter 4 Section 1
*Addition and Subtraction*
Lesson *4-9*
*Area Problems*
Lesson *4-1*
*Exponents*
Chapter 4 Section 2
*Multiplication*
Lesson *4-3*
*Multiplying Monomials*
Lesson *4-5*
*Multiplying Polynomials by Monomials*
Lesson *4-6*
*Multiplying Polynomials*
Chapter *4*
*Polynomials*
Lesson *4-4*
*Powers of Monomials*
Chapter 4 Section 3
*Problem Solving*
Lesson *4-10*
*Problems Without Solutions*
Lesson *4-8*
*Rate-Time-Distance Problems*
Lesson *4-7*
*Transforming Formulas*
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.
First power of 5:
5 to the first power = 5
Fourth power of 5:
5 to the fourth power = 5 · 5 · 5 · 5
Second power of 5:
5 to the second power = 5 · 5
Third power of 5:
5 to the third power = 5 · 5 · 5
Chapter 4 Summary 7.
A chart can be used to solve problems about distances or areas. Formulas to use are: rate × time = distance length × width = area of a rectangle
Chapter 4 Summary 6.
A formula may be transformed to express a particular variable in terms of the other variables.
*simplest form of a polynomial*
A polynomial is in simplest form when no two of its terms are similar.
*trinomial*
A polynomial of only three terms.
*binomial*
A polynomial of only two terms.
*polynomial*
A sum of monomials.
*monomial*
An expression that is either a numeral, a variable, or the product of a numeral and one or more variables.
*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* *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, you add 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.
*Chapter* 4 *Summary*
Polynomials Summary
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 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 1.
The expression *b*ⁿ is an abbreviation for *b⋅b⋅b⋅...⋅b.* (n factors) The base is *b* and the exponent is n.
*exponential form of a power*
The expression n⁴ is the exponential form of n ⋅n ⋅ n ⋅ n.
*degree of a polynomial*
The greatest of the degrees of its terms after it has been simplified.
*base of a power*
The number that is used as a factor a given number of times; in 2⁵, 2 is the base.
*degree of a variable in a monomial*
The number of times that the variable occurs as a factor in the monomial.
*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.
*degree of a monomial*
The sum of the degrees of the variables in the monomial.
Chapter 4 Summary 3.
To add (or subtract) polynomials, you add (or subtract) their similar terms. Similar terms are monomials that are exactly alike or that differ only in their numerical coefficients.
*Objective* 4-2
To add and subtract polynomials.
*Objective* 4-4
To find powers of monomials.
*Objective* 4-5
To multiply a polynomial by a monomial.
*Objective* 4-3
To multiply monomials.
*Objective* 4-6
To multiply polynomials.
*Objective* 4-10
To recognize problems that do not have solutions.
Chapter 4 Summary 2.
To simplify expressions that contain powers follow the steps used to simplify numerical expressions.
Chapter 4 Summary 8.
To solve problems involving area, you may find it helpful to make a sketch.
*Objective* 4-9
To solve some problems involving area.
*Objective* 4-8
To solve some word problems involving uniform motion.
*Objective* 4-7
To transform a formula.
*Objective* 4-1
To write and simplify expressions involving exponents.
*similar terms*
Two monomials that are exactly alike or are the same except for their numerical coefficients. Also called *like terms.*
The number 25 can be written as 5 · 5 and is called
a power of 5
Caution: Be careful when an expression contains both
parentheses and exponents.
