4.4 Real Zeros of Polynomial Functions
nth degree polynomial function: f(x)=anxn+an-1xn-1+ ... +a1x+a0.
≠0
RATIONAL ROOT THEOREM:
Let f(x)=anxn+an-1xn-1+ ... + a1x+a0 is a polynomial with integer coefficients,
then any rational zero must be of the form
where p is a factor of a0 and q is a factor of an.
Example: Find zeros of f(x)=x3-2x2-5x+6...
By the theorem, f(x) has possible rational zeros at:
_____________ .
Use synthetic division to find a zero:
Example: Find zeros of f(x)=2x3-13x2-7x+18...
By the theorem, f(x) has possible rational zeros at:
_____________ .
Use synthetic division to find a zero:
1
Factor completely by using the rational root theorem:
2x4­5x3+4x2­5x+2
Solve using the rational root theorem:
x3 = ­5x2+8x­2
2
Let f(x) = 3x4-4x3-11x2+16x-4. Find a factored form. Identify each root and its multiplicity.
What does the graph look like?
3
Let f(x) = x4+x3-3x2-5x-2. Find a factored form. Identify each root and its multiplicity.
What does the graph look like?
4
You try:
1)
Solve: 7x3­2x=10­5x2
2)
Factor completely: x4+17x3+11x2­5x
3) What is the multiplicity of 1 in f(x)=x4­3x3+3x2­x ?
5