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

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

A constant is a number whose value does not change. Constant

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. Term

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

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

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

A binomial is a polynomial consisting of exactly two terms. Binomial

A trinomial is a polynomial consisting of exactly three terms. Trinomial

Monomial: 3x Binomial: 3x + 4 Trinomial: 2x 2 + 3x + 4

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

Is 2x – 1 + 5 a polynomial? If not, why not? no; negative exponent Example

Is 2 – 1 x + 5 a polynomial? If not, why not? yes (The exponent of the variable is 1.) Example

Is a polynomial? If not, why not? no; negative exponent 3x3x 3x3x Example

Is 4y 2 x + 3x 2 y a polynomial? If not, why not? yes Example

Is 3x a polynomial? If not, why not? no; fractional exponent 1212 1212 Example

Is 3 x a polynomial? If not, why not? no; variable exponent Example

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

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

Polynomial Degree 2 2 0 0 2 is the same as 2x 0, since x 0 = 1.

Polynomial Degree 3x3x 3x3x 1 1 The variable x is the same as x 1.

Polynomial Degree – 2x2y– 2x2y – 2x2y– 2x2y 3 3 The variable y has an exponent of 1; 2 + 1 = 3.

Polynomial Degree x 3 y 2 + 3x 2 + 2y 4 – 8 5 5 The term x 3 y 2 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). 5; binomial Example

Give the degree of – 5 and identify the type of polynomial by special name (monomial, binomial, or trinomial). 0; monomial Example

Give the degree of 4x 8 + 3x – 1 and identify the type of polynomial by special name (monomial, binomial, or trinomial). 8; trinomial Example

Give the degree of 3xy 2 + 5x and identify the type of polynomial by special name (monomial, binomial, or trinomial). 3; binomial Example

Give the degree of (– 5) 2 and identify the type of polynomial by special name (monomial, binomial, or trinomial). 0; monomial Example

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

Evaluate x 2 – 6x – 18 when x = 7. = – 11 x 2 – 6x – 18 = 7 2 – 6(7) – 18 = 49 – 42 – 18 Example 1

Evaluate – 3x 3 y + 4x 2 y 2 when x = 6 and y = – 2. = 1,872 – 3x 3 y + 4x 2 y 2 = – 3(6) 3 (– 2) + 4(6) 2 (– 2) 2 = – 3(216)(– 2) + 4(36)(4) Example 2 = 1,296 + 576

Evaluate 3x 2 + 4x – 5 when x = 2. 15 Example

Evaluate 7x 2 – 4z 3 when x = – 4 and z = – 3. 220 Example

Evaluate – 3x 2 yz when x = 2, y = – 4 and z = 3. 144 Example

Sometimes a polynomial is substituted for a variable. Evaluate 4y 2 + 3y if y = 3x. 36x 2 + 9x Example

Evaluate x 5 (5 – y) when x = –, y = – 1, and z = 4. Evaluate x 5 (5 – y) when x = –, y = – 1, and z = 4. 4545 4545 96 25 Exercise

Evaluate – 6 when x = –, y = – 1, and z = 4. Evaluate – 6 when x = –, y = – 1, and z = 4. 4545 4545 3z23z2 3z23z2 00 Exercise

Evaluate x – 2y 2 when x = –, y = – 1, and z = 4. Evaluate x – 2y 2 when x = –, y = – 1, and z = 4. 4545 4545 14 5 –– Exercise

Evaluate 3z – when x = –, y = – 1, and z = 4. Evaluate 3z – when x = –, y = – 1, and z = 4. 4545 4545 2y52y5 2y52y5 62 5 Exercise

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