AP Precalculus
~ 1.5: Polynomials and Complex Zeros ~
~ 1.5: Polynomials and Complex Zeros ~
Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.
Determine if a polynomial function is even or odd.
If a is a complex number and p(a) = 0, then a is called a zero of the polynomial function p, or a root of p(x) = 0.
If a is a real number, then (x − a) is a linear factor of p if and only if a is a zero of p.
If a linear factor (x a − ) is repeated n times, the corresponding zero of the polynomial function has a multiplicity n.
A polynomial function of degree n has exactly n complex zeros when counting multiplicities.
If a is a real zero of a polynomial function p, then the graph of y = p(x) has an x-intercept at the point (a, 0).
Real zeros of a polynomial can be endpoints for intervals satisfying polynomial inequalities.
If a + bi is a non-real zero of a polynomial function p, then its conjugate a − bi is also a zero of p.
If the real zero, a, of a polynomial function has even multiplicity, then the signs of the output values are the same for input values near x = a. For these polynomial functions, the graph will be tangent to the x-axis at x = a.
The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial function is equal to the least value n for which the successive nth differences are constant.
An even function is graphically symmetric over the line x = 0 and analytically has the property f (−x) = f x( ).
If n is even, then a polynomial of the form p(x) = anxn , where n ≥ 1 and an ≠ 0, is an even function.
An odd function is graphically symmetric about the point (0,0) and analytically has the property f (−x) = − f x( ).
If n is odd, then a polynomial of the form p(x) = a xn n , where n ≥ 1 and an ≠ 0, is an odd function.
The video for AP Precalculus, 1.5: Polynomials and Complex Zeros will come out on Thursday, September 14, 2023.