An algebraic expression is a mathematical expression containing numbers, variables, and operational signs.

An algebraic expression is a mathematical expression containing numbers, variables, and operational signs.

Algebraic ExpressionAlgebraic Expression

A constant is a number whose value does not change.

A constant is a number whose value does not change.

ConstantConstant

A term is a constant, a variable, or the product of a constant and one or more variables. Terms are always separated from each other by a plus or minus sign.

A term is a constant, a variable, or the product of a constant and one or more variables. Terms are always separated from each other by a plus or minus sign.

TermTerm

The numerical coefficient is the number factor (constant) accompanying the variable in a term.

The numerical coefficient is the number factor (constant) accompanying the variable in a term.

Numerical CoefficientNumerical Coefficient

A monomial is a real number, a variable, or the product of a real number and variable(s) with nonnegative integer exponents.

A monomial is a real number, a variable, or the product of a real number and variable(s) with nonnegative integer exponents.

MonomialMonomial

A polynomial is a monomial or the sum or difference of two or more monomials.

A polynomial is a monomial or the sum or difference of two or more monomials.

PolynomialPolynomial

A binomial is a polynomial consisting of exactly two terms.

A binomial is a polynomial consisting of exactly two terms.

BinomialBinomial

A trinomial is a polynomial consisting of exactly three terms.

A trinomial is a polynomial consisting of exactly three terms.

TrinomialTrinomial

Monomial: 3x

Binomial: 3x + 4

Trinomial: 2x2 + 3x + 4

ExampleExampleIs 4x + 5 a polynomial? If not, why not?Is 4x + 5 a polynomial? If not, why not?

yesyes

Is 2x – 1 + 5 a polynomial? If

not, why not?Is 2x

– 1 + 5 a polynomial? If not, why not?

no; negative exponentno; negative exponent

ExampleExample

Is 2 – 1x + 5 a polynomial? If

not, why not?Is 2

– 1x + 5 a polynomial? If not, why not?

yes (The exponent of the variable is 1.)

yes (The exponent of the variable is 1.)

ExampleExample

Is a polynomial? If not,

why not?

Is a polynomial? If not,

why not?

no; negative exponentno; negative exponent

3x3x

ExampleExample

Is 4y2x + 3x2y a polynomial? If not, why not?Is 4y2x + 3x2y a polynomial? If not, why not?

yesyes

ExampleExample

Is 3x a polynomial? If not, why not?Is 3x a polynomial? If not, why not?

no; fractional exponentno; fractional exponent

1212

ExampleExample

Is 3x a polynomial? If not, why not?Is 3x a polynomial? If not, why not?

no; variable exponentno; variable exponent

ExampleExample

The sum of the exponents of the variables contained in a term is the degree of the term.

The sum of the exponents of the variables contained in a term is the degree of the term.

Degree of the TermDegree of the Term

The highest degree of any of the individual terms of a polynomial is the degree of the polynomial.

The highest degree of any of the individual terms of a polynomial is the degree of the polynomial.

Degree of the PolynomialDegree of the Polynomial

PolynomialPolynomial

DegreeDegree

22

00

2 is the same as 2x0, since x0 = 1.

2 is the same as 2x0, since x0 = 1.

PolynomialPolynomial

DegreeDegree

3x3x

11

The variable x is the same as x1.

The variable x is the same as x1.

PolynomialPolynomial

DegreeDegree

– 2x2y– 2x2y

33

The variable y has an exponent of 1; 2 + 1 = 3.

The variable y has an exponent of 1; 2 + 1 = 3.

PolynomialPolynomial

DegreeDegree

x3y2 + 3x2 + 2y 4 – 8x3y2 + 3x2 + 2y 4 – 8

55

The term x3y2 has a degree of 5. This is the highest-

degree term in the polynomial.

The term x3y2 has a degree of 5. This is the highest-

degree term in the polynomial.

Give the degree of 3x 5 + 1

and identify the type of polynomial by special name (monomial, binomial, or trinomial).

Give the degree of 3x 5 + 1

and identify the type of polynomial by special name (monomial, binomial, or trinomial).

5; binomial5; binomial

ExampleExample

Give the degree of – 5 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

Give the degree of – 5 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

0; monomial0; monomial

ExampleExample

Give the degree of 4x8 + 3x – 1 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

Give the degree of 4x8 + 3x – 1 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

8; trinomial8; trinomial

ExampleExample

Give the degree of 3xy2 + 5x and identify the type of polynomial by special name (monomial, binomial, or trinomial).

Give the degree of 3xy2 + 5x and identify the type of polynomial by special name (monomial, binomial, or trinomial).

3; binomial3; binomial

ExampleExample

Give the degree of (– 5)2 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

Give the degree of (– 5)2 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

0; monomial0; monomial

ExampleExample

Give the degree of (– x)2 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

Give the degree of (– x)2 and identify the type of polynomial by special name (monomial, binomial, or trinomial).

2; monomial2; monomial

ExampleExample

Evaluate x2 – 6x – 18 when x = 7.Evaluate x2 – 6x – 18 when x = 7.

= – 11= – 11

x2 – 6x – 18x2 – 6x – 18 = 72 – 6(7) – 18= 72 – 6(7) – 18= 49 – 42 – 18= 49 – 42 – 18

Example 1Example 1

Evaluate – 3x3y + 4x2y2

when x = 6 and y = – 2.Evaluate – 3x3y + 4x2y2

when x = 6 and y = – 2.

= 1,872= 1,872

– 3x3y + 4x2y2– 3x3y + 4x2y2

= – 3(6)3(– 2) + 4(6)2(– 2)2= – 3(6)3(– 2) + 4(6)2(– 2)2

= – 3(216)(– 2) + 4(36)(4)= – 3(216)(– 2) + 4(36)(4)

Example 2Example 2

= 1,296 + 576= 1,296 + 576

Evaluate 3x2 + 4x – 5 when x = 2.Evaluate 3x2 + 4x – 5 when x = 2.

1515

ExampleExample

Evaluate 7x2 – 4z3 when x = – 4 and z = – 3.Evaluate 7x2 – 4z3 when x = – 4 and z = – 3.

220220

ExampleExample

Evaluate – 3x2yz when x = 2, y = – 4 and z = 3.Evaluate – 3x2yz when x = 2, y = – 4 and z = 3.

144144

ExampleExample

Sometimes a polynomial is substituted for a variable. Evaluate 4y2 + 3y if y = 3x.

Sometimes a polynomial is substituted for a variable. Evaluate 4y2 + 3y if y = 3x.

36x2 + 9x36x2 + 9x

ExampleExample

Evaluate x5(5 – y) when

x = – , y = – 1, and z = 4.

Evaluate x5(5 – y) when

x = – , y = – 1, and z = 4.4545

96259625

ExerciseExercise

Evaluate – 6 when

x = – , y = – 1, and z = 4.

Evaluate – 6 when

x = – , y = – 1, and z = 4.4545

3z23z2

00

ExerciseExercise

Evaluate x – 2y2 when

x = – , y = – 1, and z = 4.

Evaluate x – 2y2 when

x = – , y = – 1, and z = 4.4545

145

145

––

ExerciseExercise

Evaluate 3z – when

x = – , y = – 1, and z = 4.

Evaluate 3z – when

x = – , y = – 1, and z = 4.4545

2y5

2y5

625

625

ExerciseExercise