Download 2.5 Zeros of Polynomial Functions

2.5 Zeros of Polynomial Functions
We have known that a polynomial of degree n can have up to n zeros.
Working with the complex number system now allows us to state
that it has exactly n zeros when taking into account multiplicity.
Find all the zeros of each function:
f(x) = x - 4
f(x) = x2 - 4x + 4 = (x - 2)(x - 2)
f(x) = x(x + 3i)(x - 3i)
f(x) = (x + 2)(x - 2)(x + 2i)(x - 2i)
The Rational Zero Test relates the possible rational zeros of a
polynomial to the leading coefficient and to the constant term.
The Rational Zero Test:
If a polynomial f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 has integer
coefficient, every rational zero of f has the form p/q, where p and q
have no common factors other than 1, and p is a factor of the
constant term a0 and q is a factor of the leader coefficient an.
Make a list of the possible rational zeros:
f(x) = x3 - 7x - 6
f(x) = 4x3 - 19x2 + 33x - 9
1
Find all the rational zeros of f(x) = x3 + x + 1
Find all the rational zeros of f(x) = x3 - 5x2 + 2x + 8
f(x) = x3 - 15x2 + 75x - 125
f(x) = 2x4 - 9x3 - 18x2 + 71x - 30
2
Find the zeros of f(x) = 6x4 - 11x3 - 51x2 + 99x - 27
Complex Zeros Occur in Conjugate Pairs:
If a + bi, where b ≠ 0, is a zero of a function, then a - bi is also a zero.
Find a polynomial function that has 2, 7i, and -7i as zeros:
Find a polynomial function that has 6, -5 + 2i, and -5 - 2i as zeros:
3
Find all the zeros of f(x) = 2x3 + 3x2 + 50x + 75 if 5i is one zero:
Find all the zeros of f(x) = x3 - 4x2 + 21x - 34 if 1 + 4i is one zero:
Find the zeros and write as a product of linear factors:
g(x) = x2 + 10x + 23
h(x) = x3 - 11x2 + 41x - 51
4
Descartes's Rule of Signs:
1) The number of positive real zeros of a function f is either equal
to the number of variations in sign of f(x) or less than that number
by an even integer.
2) The number of negative real zeros of a function f is either equal
to the number of variations in sign of f(-x) or less than that number
by an even integer.
A variation in sign means that two consecutive coefficients have
opposite signs. Count your multiplicities as multiple zeros.
f(x) = -2x3 + 5x2 - x + 8
A value is called an upper bound of a function if no zeros are greater
than that value. Similarly with a lower bound.
Upper and Lower Bound Rules:
Let f(x) be a polynomial with real coefficients and a positive leading
coefficient. Suppose f(x) is divided by x - c, using synthetic division:
1) If c > 0 and each number in the last row is either positive or zero,
c is an upper bound of f.
2) If c < 0 and the numbers in the last row are alternately positive
and negative (zero counts as either), c is a lower bound of f.
Verify an upper bound of 4 and a lower bound of -1 for 2x4 - 8x + 3
5
Find all the real zeros of f(x) = 6x3 - 4x2 + 3x - 2
Find all the real zeros of f(x) = 8x3 - 4x2 + 6x - 3
6