Polynomials. Characteristics of Polynomials DEFINITION: an algebraic expression consisting of two or more terms (n ≥ 2). 1. Usually has one variable (x)

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Presentation transcript:

Polynomials

Characteristics of Polynomials DEFINITION: an algebraic expression consisting of two or more terms (n ≥ 2). 1. Usually has one variable (x) 2. Variable is raised to a non-negative power (x^n>0) 3. multiplied by a constant

degree of polynomial DEFINITION: the greatest power to which the variable is raised Example: The degree of this trinomial is 3

CUBIC FUNCTIONS n = 3 (three roots) one or three of these roots will be real numbers (the others will be complex numbers) cubic functions with three roots that are real numbers will have 3 x-intercepts; y = (x - 5) if a>0, the cubic function will start in the third quadrant if a<0, the cubic function will start in the second quadrant complex numbers involve the square root of a negative number; and is not a real number.

Approximately graph the following y=(x-1)(x-2)(x+3) = x^3 - 7x + 6 Think: - From which quadrant does the function begin? (I, II, III, IV) - How many x- intercepts are there? At what point do they cross the horizontal?

Approximately graph the following y = (x - 2)^3 = x^3 - 6x^2 + 12x - 8 Think: - From which quadrant does the function begin? (I, II, III, IV) - How many x- intercepts are there? At what point do they cross the horizontal?

Two other types of Cubic Functions One real root and two complex roots

Two other types of Cubic functions Two equal real roots and one other real root

BELLRINGER What is the equation for this cubic function?

QUARTIC FUNCTIONS n = 4 Either four roots, two roots or no roots are real. Non-real roots are complex. If starting in quadrant 3, the function will end in quadrant 4 (start at the bottom, leave at the bottom); a < 0 If starting in quadrant 2, the function will end in quadrant 1 (start at the top, leave at the top); a > 0 Quartic Functions have 3 turns unless all 4 real roots are equal

Number of x-intercepts How many x-intercepts are there in this function? 4 That means the equation has four different real roots Is "a" positive or negative? Because this function starts in Quadrant II, we know that "a" is > 0 The x-intercept rules for quartic functions is the same as cubic functions. 2 real roots that are equal results in the line kissing the horizontal at its turn

graph this function: y = x^2(x^2 - 4)