5.6 Find Rational Zeros
Goal  Find all real zeros of a polynomial function.
Your Notes
THE RATIONAL ZERO THEOREM
If f(x) = anxn + …+ a1x + a0 has _integer_ coefficients, then every rational zero of f has
the following form:
p
factor of constant term a0

q
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
1
2
3
10
is _10_. So, the possible rational zeros are: x =  ,  ,  , 
1
1
1
1
2. Test these zeros using synthetic division. Test x = _1_:
1 4 7
10
1
3
 10
1
1
3
 10
0
__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_.
Checkpoint Find all real zeros of the function.
1.
f(x) = x3 + 3x2  l0x  24
2, 3, and 4
Your Notes
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:
1 3 1 3 1 3 1 3
 , , , , , , ,
1 1 2 2 4 4 8 8
2. Choose reasonable values using the function's graph.
3
1
1
3
x=  ,x ,x ,x
2
2
4
2
3. Check the chosen values using synthetic division.

3
2
8
8
2
 12
 21
7
15
9
 10
6
3
3
2
0

3
2
is a zero.
4. Factor out a binomial.
3 3

2
f(x) =  x  (8 x  10 x  6 x  2)
2


=x 

3
(2)(4 x 3  5 x 2  3x  1)
2
= _(2x + 3)(4x3  5x2  3x + 1)_
Write as a product of factors.
Factor out _2_.
Multiply by _2_.
5. Repeat the steps above for g(x) = _4x3  5x2  3x+ 1_. Any zero of g will also be a
1
zero of f. Synthetic division shows that
is a zero and yields the quotient
4
____
_4x2  4x 4_. Factoring a 4 out of the quotient yields
f(x) = _(2x+ 3)(4x  1)(x2  x  1)_.
6. Find the remaining zeros by solving _(x2  x 1)_ = 0.
1  (1) 2  4(1)(1)
Use quadratic formula.
2
1 5
x=
Simplify.
2
3 1 1 5
1 5
The real zeros of f are  , ,
, and,
2 4
2
2 .
x=
Your Notes
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_
3
Write the equation.
2
16 = _x + 6x + 9x_
Multiply.
0 = _x3 + 6x2 + 9x  16_
Subtract _16_ from each side.
Find the possible rational solutions:
1
2
4
8 16 .
 ,  ,  ,  ,
1
1
1
1
1
Test the possible solutions. Only positive x-values make sense.
1
1
1
6
1
7
9
7
16
 16
16
0
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.
Checkpoint Find all real zeros of the function.
2. f(x) = 9x4 + 12x3  25x2  11x + 6
2 1 1 13
1 13
 , ,
, and
3 3
2
2
Homework
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