Identify terms and coefficients. Know the vocabulary for polynomials. Add like terms. Add and subtract polynomials. Evaluate polynomials Unit – Polynomials

Identify terms and coefficients. In an expression such as the quantities 4x 3, 6x 2, 5x, and 8 are called terms. In the first term 4x 3, the number 4 is called the coefficient, of x 3. In the same way, 6 is the coefficient of x 2 in the term 6x 2, and 5 is the coefficient of x in the term 5x. The constant term is 8. Slide 5.4-4

Solution: Name the coefficient of each term in the expression Slide EXAMPLE 1 Identifying Coefficients

Know the vocabulary for polynomials. A polynomial in x is a term or the sum of a finite number of terms of the form ax n, for any real number a and any whole number (no negative, no fraction) n. For example, is a polynomial in x. (The 4 can be written as 4x 0.) This polynomial is written in standard form, since the exponents on x decrease from left to right. By contrast, is not a polynomial in x, since x appears in a denominator. Polynomial in x Not a Polynomial Slide A polynomial can be defined using any variable and not just x. In fact, polynomials may have terms with more than one variable.

The degree of a term is the sum of the exponents on the variables. For example 3x 4 has degree 4, while the term 5x (or 5x 1 ) has degree 1, −7 has degree 0 ( since −7 can be written −7x 0 ). The degree of a polynomial is the greatest degree term of the polynomial. For example 3x 4 + 5x is of degree 4. Slide Know the vocabulary for polynomials. (cont’d)

Three types of polynomials are common and given special names. A polynomial with only one term is called a monomial. (Mono- means “one,” as in monorail.) Examples are and Monomials A polynomial with exactly two terms is called a binomial. (Bi- means “two,” as in bicycle.) Examples are and Binomials A polynomial with exactly three terms is called a trinomial. (Tri- means “three,” as in triangle.) Examples are and Trinomials Slide Know the vocabulary for polynomials. (cont’d)

DegreeNameExample 0Constant5 1Linear2xy + 4 2Quadratic4x 2 – 7x + 2 3Cubicx 3 – 2x 2 – 2x + 1 4Quartic-8x 4 – 7x 2 + 3x - 4 5Quintic3x 5 – 5x 4 + 2x 3 – 4x 2 +10y Number of TermsNameExample 1Monomial3x 2 2Binomial5x + 4x 3 3Trinomial3x + 4x More than 3Polynomial5x 5 + 4x 4 - 2x 3 + 8x 2 - x - 1 Know the vocabulary for polynomials. (cont’d)

Solution: Write polynomial in standard form, give the degree, and tell whether the polynomial is a monomial, binomial, trinomial. Slide EXAMPLE 3 Classifying Polynomials 3x + 5x x 3 + 3x – 4 Degree 3 or cubic, trinomial

Add like terms. like terms have exactly the same combinations of variables, with the same exponents on the variables. Only the coefficients may differ. We combine, or add, like terms by adding their coefficients. Examples of like terms Slide 5.4-7

Solution: Simplify by adding like terms. Unlike terms cannot be combined. Unlike terms have different variables or different exponents on the same variables. Slide EXAMPLE 2 Adding Like Terms

Add and subtract polynomials. Polynomials may be added, subtracted, multiplied, and divided. Subtracting Polynomials To subtract two polynomials, change all the signs in the second polynomial and add the result to the first polynomial. Adding Polynomials To add two polynomials, add like terms. Slide

Add. and Solution: + + Slide EXAMPLE 5 Adding Polynomials Vertically

Solution: Add. Slide EXAMPLE 6 Adding Polynomials Horizontally

Perform the subtractions. from Solution: Slide EXAMPLE 7 Subtracting Polynomials Horizontally

Subtract. Solution : + Slide EXAMPLE 8 Subtracting Polynomials Vertically

Subtract. Solution: Slide EXAMPLE 9 Adding and Subtracting Polynomials with More Than One Variable

Solution: Find the value of 2y 3 + 8y − 6 when y = −1. Use parentheses around the numbers that are being substituted for the variable, particularly when substituting a negative number for a variable that is raised to a power, or a sign error may result. Slide EXAMPLE 4 Evaluating a Polynomial