5.6 Find Rational Zeros

 

THE RATIONAL ZERO THEOREM

If f(x) = anxn + …+ a1x + a0 has _integer_ coefficients, then every rational zero of f has the following form:

   factor of constant term       a0

   factor of leading coefficient         an

 

Example 1

Find zeros when the leading coefficient is 1

 

Find all real zeros of f(x) = x3 - 4x2 - 7x + 10.

 

1.      List the possible rational zeros. The leading coefficient is _1_ and the constant term is _10_. So, the possible rational zeros are: x =

2.     Test these zeros using synthetic division. Test x = _1_:

__1__ is a zero.

3.     Factor the trinomial and use the factor theorem.

f(x) = (_x- 1_)(x2 - 3x - 10_).

 = _(x- 1)(x2 + 2)(x - 5)_.

The zeros of f are _1, -2, and 5_.

 

Example 2

Find zeros when the leading coefficient is not 1

 

Find all real zeros of f(x) = 8x4 + 2x3 - 21x2 - 7x + 3.

1.  List the possible rational zeros of f:

 


2.  Choose reasonable values using the function's graph.

x =                                             

 

 

Check the chosen values using synthetic division.

is a zero.

3.   Factor out a binomial.

f(x) =

Write as a product of factors.

=

Factor out _2_.

= _(2x + 3)(4x3 - 5x2 - 3x + 1)_

Multiply by _2_.

4.    Repeat the steps above for g(x) = _4x3 - 5x2 - 3x+ 1_. Any zero of g will also be a zero of f. Synthetic division shows that ____is a zero and yields the quotient

_4x2 - 4x- 4_. Factoring a 4 out of the quotient yields

f(x) = _(2x+ 3)(4x - 1)(x2 - x - 1)_.

Find the remaining zeros by solving _(x2 - x- 1)_ = 0.

x =

Use quadratic formula.

x =

Simplify.

The real zeros of f are                                 and,               .

Example 3

Solve a multi-step problem

 

Sandbox You are building a wooden square sandbox for a local playground. You want the volume of the box to be 16 cubic feet. You want the height of the box to be x feet and the length of each side of the square base to be x + 3 feet. What are the dimensions?

 

 

 

 

Solution

 

The volume is V = Bh where B = base area and h = height.

Volume

(cubic feet)

=

Area of base

(square feet)

·

Height (feet)

 

16           =        (x + 3)2        ·            x

16 = _(x + 3)2 x_

Write the equation.

16 = _x3 + 6x2 + 9x_

Multiply.

0 = _x3 + 6x2 + 9x - 16_

Subtract _16_ from each side.

 

Find the possible rational solutions:

.

Test the possible solutions. Only positive x-values make sense.

1

Check for other solutions. The other possible rational solutions _are not_ solutions, so
x = _1_ is the solution. The height of the sandbox should be _1_ foot and each side of the base should be _1_+ 3 = _4_ feet.