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Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
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Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Percents
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
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Simplifying Complex Fractions
Algebra
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
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Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
fractions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Quadratics
Operations on Radicals
Factoring a Difference of Two Squares
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Solving Quadratic Equations by Factoring
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Simplifying Expressions Involving Variables
Adding Integers
Decimals
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
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Polynomials

A term is an expression containing a number or the product of a number and one or more variables raised to powers. Some examples of terms are

4x3, -x2y3, 6ab, and -2.

A polynomial is a single term or a finite sum of terms. The powers of the variables in a polynomial must be positive integers. For example,

4x3 + (-15x2) + x + (-2)

is a polynomial. Because it is simpler to write addition of a negative as subtraction, this polynomial is usually written as

4x3 -15x2 + x -2

The degree of a polynomial in one variable is the highest power of the variable in the polynomial. So 4x3 -15x2 + x -2 has degree 3 and 7w -w2 has degree 2.

The degree of a term is the power of the variable in the term. Because the last term has no variable, its degree is 0.

4x3  -15x2 + x  -2
Third-degree term Second-degree term First-degree term Zero-degree term

A single number is called a constant and so the last term is the constant term. The degree of a polynomial consisting of a single number such as 8 is 0.

The number preceding the variable in each term is called the coefficient of a variable or the coefficient of that term. In 4x3 -15x2 + x -2 the coefficient of x3 is 4, the coefficient of x2 is -15, and the coefficient of x is 1 because x = 1 · x.

 

Identifying coefficients

Determine the coefficients of x3 and x2 in each polynomial:

a) x3 + 5x2 - 6

b) 4x6 - x3 + x

Solution

a) Write the polynomial as 1 · x3 + 5x2 - 6 to see that the coefficient of x3 is 1 and the coefficient of x2 is 5.

b) The x2-term is missing in 4x6 - x3 + x. Because 4x6 - x3 + x can be written as

4x6 - 1 · x3 + 0 · 4x6 - x2 + x,

the coefficient of x3 is -1 and the coefficient of xis 0.

For simplicity we generally write polynomials with the exponents decreasing from left to right and the constant term last. So we write

x3 - 4x2 + 5x + 1 rather than  - 4x2 + 1 + 5x + x3

When a polynomial is written with decreasing exponents, the coefficient of the first term is called leading coefficient.

Certain polynomials are given special names. A monomial is a polynomial that has one term, a binomial is a polynomial that has two terms, and a trinomial is a polynomial that has three terms. For example, 3x5 is a monomial, 2x - 1 is a binomial, and 4x6 -3x + 2 is a trinomial.

 

Types of polynomials

Identify each polynomial as a monomial, binomial, or trinomial and state its degree.

a) 5x2 - 7x3 + 2

b) x43 - x2

c) 5x

d) -12

Solution

a) The polynomial 5x2 - 7x3 + 2 is a third-degree trinomial.

b) The polynomial x43 - x2 is a binomial with degree 43.

c) Because 5x = 5x1, this polynomial is a monomial with degree 1.

c) The polynomial -12 is a monomial with degree 0.

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