5.8Rational Zero Theorem

THE RATIONAL ZERO THEOREM:

If f(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x1 + a0

has integer coefficients, then every rational zero of f(x) has the following form:

p

q

factor of constant term a

factor of leading coefficient a0

n

FIND ALL RATIONAL ROOTS:

1. List all possible rational roots:

2. Use calculator to decide which roots to test. (Correct if remainder = 0)3. Use synthetic division to find the unknown factor4. Repeat step 2 with this new factor until you have

a quadratic.5. Factor the quadratic or use the quadratic

formula.6. Set unsolved factors to zero and solve

q

p

1. x3 – 4x2 – 11x + 30

List all possible rational roots:

Use calculator to decide which to test:

Use synthetic division to get the polynomial down to a quadratic.

Factor/Quad Formula and solveX = -3, 2, 5

EXAMPLES FIND ALL ZEROS OF THE POLYNOMIAL

EXAMPLES FIND ALL ZEROS OF THE POLYNOMIAL

2. f(x) = x3 – x2 – 6x + 8

EXAMPLES FIND ALL ZEROS OF THE POLYNOMIAL

3. f(x) = 15x4 – 68x3 – 7x2 + 24x – 4

EXAMPLES FIND ALL ZEROS OF THE POLYNOMIAL

EXAMPLES FIND ALL ZEROS OF THE POLYNOMIAL

4. f(x) = x3 – 7x2 + 10x + 6

EXAMPLE 5

The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height, and the length is 4 feet more than the height. Find the dimensions of the solid.

EXAMPLE 6

Some ice sculptures are made by filling a mold with water and then freezing it. You are making such an ice sculpture for a school dance. It is to be shaped like a pyramid with a height that is 1 foot greater than the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the dimensions of the mold?

EXAMPLE 7

You are building a solid concrete wheelchair ramp. The width of the ramp is three times the height, and the length is 5 feet more that 10 times the height. If 150 cubic feet of concrete is used, what are the dimensions of the ramp?