Dr. Claude S. Moore Danville Community College PRECALCULUS I Quadratic Functions Dr. Claude S. Moore Danville Community College
Polynomial Function A polynomial function of degree n is where the a’s are real numbers and the n’s are nonnegative integers and an 0.
a, b, and c are real numbers and a 0. Quadratic Function A polynomial function of degree 2 is called a quadratic function. It is of the form a, b, and c are real numbers and a 0.
Axis of Symmetry For a quadratic function of the form gives the axis of symmetry.
axis of symmetry: x = h vertex: (h, k) Standard Form A quadratic function of the form is in standard form. axis of symmetry: x = h vertex: (h, k)
Characteristics of Parabola vertex: maximum vertex: minimum a < 0
PRECALCULUS I Higher Degree Polynomial Functions Dr. Claude S. Moore Danville Community College
Characteristics The graph of a polynomial function… 1. Is continuous. 2. Has smooth, rounded turns. 3. For n even, both sides go same way. 4. For n odd, sides go opposite way. 5. For a > 0, right side goes up. 6. For a < 0, right side goes down.
Leading Coefficient Test: n odd graphs of a polynomial function for n odd: . an > 0 an < 0
Leading Coefficient Test: n even graphs of a polynomial function for n even: . an > 0 an < 0
Roots, Zeros, Solutions 1. x = a is root or zero of f. The following statements are equivalent for real number a and polynomial function f : 1. x = a is root or zero of f. 2. x = a is solution of f (x) = 0. 3. (x - a) is factor of f (x). 4. (a, 0) is x-intercept of graph of f (x).
Repeated Roots (Zeros) 1. If a polynomial function contains a factor (x - a)k, then x = a is a repeated root of multiplicity k. 2. If k is even, the graph touches (not crosses) the x-axis at x = a. 3. If k is odd, the graph crosses the x-axis at x = a.
Intermediate Value Theorem If a < b are two real numbers and f (x)is a polynomial function with f (a) f (b), then f (x) takes on every real number value between f (a) and f (b) for a x b.
NOTE to Intermediate Value Let f (x) be a polynomial function and a < b be two real numbers. If f (a) and f (b) have opposite signs (one positive and one negative), then f (x) = 0 for a < x < b.
Dr. Claude S. Moore Danville Community College PRECALCULUS I Polynomial and Synthetic Division Dr. Claude S. Moore Danville Community College
Full Division Algorithm If f (x) and d(x) are polynomials with d(x) 0 and the degree of d(x) is less than or equal to the degree of f(x), then q(x) and r (x) are unique polynomials such that f (x) = d(x) ·q(x) + r (x) where r (x) = 0 or has a degree less than d(x).
Short Division Algorithm f (x) = d(x) ·q(x) + r (x) dividend quotient divisor remainder where r (x) = 0 or has a degree less than d(x).
ax3 + bx2 + cx + d divided by x - k Synthetic Division ax3 + bx2 + cx + d divided by x - k k a b c d ka a r coefficients of quotient remainder 1. Copy leading coefficient. 2. Multiply diagonally. 3. Add vertically.
the remainder is r = f (k). Remainder Theorem If a polynomial f (x) is divided by x - k, the remainder is r = f (k).
Factor Theorem A polynomial f (x) has a factor (x - k) if and only if f (k) = 0.
PRECALCULUS I Real Zeros of Polynomial Functions Dr. Claude S. Moore Danville Community College
Descartes’s Rule of Signs a’s are real numbers, an 0, and a0 0. 1. Number of positive real zeros of f equals number of variations in sign of f(x), or less than that number by an even integer. 2. Number of negative real zeros of f equals number of variations in sign of f(-x), or less than that number by an even integer.
Example 1: Descartes’s Rule of Signs a’s are real numbers, an 0, and a0 0. 1. f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots. 2. f(-x) = -4x3 - 5x2 + 6 has one change-of-signs; thus, f(x) has one negative real root.
Example 2: Descartes’s Rule of Signs Factor out x; f(x) = x(4x2 - 5x + 6) = xg(x) 1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots. 2. g(-x) = 4x2 + 5x + 6 has zero change-of-signs; thus, g(x) has no negative real root.
Rational Zero Test If a’s are integers, every rational zero of f has the form rational zero = p/q, in reduced form, and p and q are factors of a0 and an, respectively.
Example 3: Rational Zero Test f(x) = 4x3 - 5x2 + 6 p {1, 2, 3, 6} q {1, 2, 4} p/q {1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4} represents all possible rational roots of f(x) = 4x3 - 5x2 + 6 .
Upper and Lower Bound f(x) is a polynomial with real coefficients and an > 0 with f(x) (x - c), using synthetic division: 1. If c > 0 and each # in last row is either positive or zero, c is an upper bound. 2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound.
Example 4: Upper and Lower Bound 2x3 - 3x2 - 12x + 8 divided by x + 3 -3 2 -3 -12 8 -6 27 -45 2 -9 15 -37 c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a lower bound to real roots.