Polynomials and Polynomial Functions 5.3 Polynomials and Polynomial Functions

Polynomial Vocabulary Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial – a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

Polynomial Vocabulary In the polynomial 7x5 + x2y2 – 4xy + 7 There are 4 terms: 7x5, x2y2, -4xy and 7. The coefficient of term 7x5 is 7, of term x2y2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term.

Types of Polynomials Monomial is a polynomial with one term. Binomial is a polynomial with two terms. Trinomial is a polynomial with three terms.

Degrees Degree of a term The degree of a term is the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a4b3c is 8 (remember that c can be written as c1).

Degrees Degree of a polynomial The degree of a polynomial is the greatest degree of all its terms. Degree of 9x3 – 4x2 + 7 is 3.

Evaluating Polynomials Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Example: Find the value of 2x3 – 3x + 4 when x = 2. 2x3 – 3x + 4 = 2(2)3 – 3(2) + 4 = 2(8) + 6 + 4 = 6

Combining Like Terms Example: Like terms are terms that contain exactly the same variables raised to exactly the same powers. Warning! Only like terms can be combined through addition and subtraction. Example: Combine like terms to simplify. x2y + xy – y + 10x2y – 2y + xy = x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together) = (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y

Adding Polynomials To Add Polynomials To add polynomials, combine all like terms.

Example Add: (3x – 8) + (4x2 – 3x + 3). (3x – 8) + (4x2 – 3x + 3)

Example Add: using a vertical format.

Subtracting Polynomials To Subtract Polynomials To subtract a polynomial, add its opposite.

Example Subtract 4 – (– y – 4). 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4

Example Subtract (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7).

Example Subtract:

Example Subtract:

Types of Polynomials a > 0 a < 0 Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function. A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses. A polynomial function of degree 2 is a quadratic function. In general, for the quadratic equation of the form y = ax2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0. a > 0 x a < 0 x

Types of Polynomials Polynomial functions of degree 3 are cubic functions. Cubic functions have four different forms, depending on the coefficient of the x3 term. x3 coefficient is negative x3 coefficient is positive