Fundamental
Theorem
of
3-6
3-6 Fundamental Theorem of Algebra
Algebra
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 2Algebra
Algebra22
Holt
McDougal
3-6 Fundamental Theorem of Algebra
Warm Up
Identify all the real roots of each equation.
1. 4x5 – 8x4 – 32x3 = 0
2. x3 –x2 + 9 = 9x
1, –3, 3
3. x4 + 16 = 17x2
3
–1, 1, –4, 4
2
4. 3x + 75x = 30x
Holt McDougal Algebra 2
0, –2, 4
0, 5
3-6 Fundamental Theorem of Algebra
Objectives
Use the Fundamental Theorem of
Algebra and its corollary to write a
polynomial equation of least degree
with given roots.
Identify all of the roots of a polynomial
equation.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Essential Question
• How do you write a polynomial equation of
least degree with given roots?
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
You have learned several important properties
about real roots of polynomial equations.
You can use this information to write polynomial
function when given in zeros.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Example 1: Writing Polynomial Functions
Write the simplest polynomial with roots –1, 2
,
3
and 4.
P(x) = (x + 1)(3x – 2)(x – 4)
2
P(x) = (3x + 1x – 2)(x – 4)
P(x) = 3x3 – 11x2 – 6x + 8
Holt McDougal Algebra 2
If r is a zero of P(x), then
x – r is a factor of P(x).
Multiply the first two binomials.
Multiply the trinomial by the
binomial.
3-6 Fundamental Theorem of Algebra
Check It Out! Example 1a
Write the simplest polynomial function with
the given zeros.
–2, 2, 4
P(x) = (x + 2)(x – 2)(x – 4) If r is a zero of P(x), then
x – r is a factor of P(x).
P(x) = (x2 – 4)(x – 4)
Multiply the first two binomials.
P(x) = x3– 4x2– 4x + 16
Multiply the trinomial by the
binomial.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Check It Out! Example 1b
Write the simplest polynomial function with
the given zeros.
0,
2
3
,3
If r is a zero of P(x), then
P(x) = (x – 0)(3x – 2)(x – 3)
x – r is a factor of P(x).
2
P(x) = (3x – 2x)(x – 3)
Multiply the first two binomials.
P(x) = 3x3– 11x2+ 6x
Multiply the trinomial by the
binomial.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Notice that the degree of the functions in the
previous examples is the same as the number of
zeros. This is true for all polynomial functions.
However, all of the zeros are not necessarily real
zeros. Polynomial functions, like quadratic
functions, may have complex zeros that are not
real numbers.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Using this theorem, you can write any polynomial
function in factor form.
To find all roots of a polynomial equation, you can
use a combination of the Rational Root Theorem,
the Irrational Root Theorem, and methods for
finding complex roots, such as the quadratic
formula.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Example 2: Finding All Roots of a Polynomial
Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by
finding all roots.
The polynomial is of degree 4, so there are exactly
four roots for the equation.
Step 1 Use the rational Root Theorem to identify
rational roots.
±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
Holt McDougal Algebra 2
p = –36, and q = 1.
3-6 Fundamental Theorem of Algebra
Example 2 Continued
Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find
the real roots.
Find the real roots at
or near –1 and 4.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Example 2 Continued
Step 3 Test the possible real roots.
–1
–3
–1
4
–9
36
1 –4
9
–36
0
1
Holt McDougal Algebra 2
5
–27 –36
Test –1. The remainder is
0, so (x + 1) is a factor.
3-6 Fundamental Theorem of Algebra
Example 2 Continued
3
2
The polynomial factors into (x + 1)(x – 4x + 9x – 36) = 0.
4
1
–4
4
9
0
–36
36
1
0
9
0
Holt McDougal Algebra 2
Test 4 in the cubic
polynomial. The remainder
is 0, so (x – 4) is a
factor.
3-6 Fundamental Theorem of Algebra
Example 2 Continued
2
The polynomial factors into (x + 1)(x – 4)(x + 9) = 0.
Step 4 Solve x2 + 9 = 0 to find the remaining roots.
x2 + 9 = 0
x2 = –9
x = ±3i
The fully factored form of the equation is
(x + 1)(x – 4)(x + 3i)(x – 3i) = 0. The solutions
are 4, –1, 3i, –3i.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Check It Out! Example 2
4
3
2
Solve x + 4x – x +16x – 20 = 0 by finding all
roots.
The polynomial is of degree 4, so there are exactly
four roots for the equation.
Step 1 Use the rational Root Theorem to identify
rational roots.
±1, ±2, ±4, ±5, ±10, ±20
Holt McDougal Algebra 2
p = –20, and q = 1.
3-6 Fundamental Theorem of Algebra
Check It Out! Example 2 Continued
Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find
the real roots.
Find the real roots at
or near –5 and 1.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Check It Out! Example 2 Continued
Step 3 Test the possible real roots.
–5
1
4 –1 16 –20
–5
5 –20 20
1 –1
Holt McDougal Algebra 2
4
–4
0
Test –5. The remainder is
0, so (x + 5) is a factor.
3-6 Fundamental Theorem of Algebra
Check It Out! Example 2 Continued
3
2
The polynomial factors into (x + 5)(x – x + 4x – 4) = 0.
1
1 –1
4
–4
1
0
4
0
4
0
1
Holt McDougal Algebra 2
Test 1 in the cubic polynomial.
The remainder is 0, so (x – 1)
is a factor.
3-6 Fundamental Theorem of Algebra
Check It Out! Example 2 Continued
2
The polynomial factors into (x + 5)(x – 1)(x + 4) = 0.
Step 4 Solve x2 + 4 = 0 to find the remaining roots.
x2 + 4 = 0
x2 = –4
x = ±2i
The fully factored form of the equation is (x + 5)
(x – 1)(x + 2i)(x – 2i) = 0. The solutions are –5,
1, –2i, +2i).
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Example 3: Writing a Polynomial Function with
Complex Zeros
Write the simplest function with zeros 2 + i,
and 1.
Step 1 Identify all roots.
By the Rational Root Theorem and the Complex
Conjugate Root Theorem, the irrational roots and
complex come in conjugate pairs. There are five
roots: 2 + i, 2 – i,
,
, and 1. The polynomial
must have degree 5.
Holt McDougal Algebra 2
,
3-6 Fundamental Theorem of Algebra
Example 3 Continued
Step 2 Write the equation in factored form.
P(x) = [x – (2 + i)][x – (2 – i)](x –
)[(x – (
)](x – 1)
Step 3 Multiply.
P(x) = (x2 – 4x + 5)(x2 – 3)(x – 1)
= (x4 – 4x3+ 2x2 + 12x – 15)(x – 1)
P(x) = x5 – 5x4+ 6x3 + 10x2 – 27x + 15
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Check It Out! Example 3
Write the simplest function with zeros 2i, 1+
and 3.
Step 1 Identify all roots.
By the Rational Root Theorem and the Complex
Conjugate Root Theorem, the irrational roots and
complex come in conjugate pairs. There are five
roots: 2i, –2i,
,
, and 3. The polynomial
must have degree 5.
Holt McDougal Algebra 2
2,
3-6 Fundamental Theorem of Algebra
Check It Out! Example 3 Continued
Step 2 Write the equation in factored form.
P(x) = [ x - (2i)][x + (2i)][x - ( 1 + x )][x - (1 - x )](x - 3)
Step 3 Multiply.
P(x) = x5 – 5x4+ 9x3 – 17x2 + 20x + 12
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Lesson Quiz: Part I
Write the simplest polynomial function with
the given zeros.
1. 2, –1, 1
x3 – 2x2 – x + 2
2. 0, –2,
x4 + 2x3 – 3x2 – 6x
3. 2i, 1, –2
x4 + x3 + 2x2 + 4x – 8
4. Solve by finding all roots.
x4 – 5x3 + 7x2 – 5x + 6 = 0
Holt McDougal Algebra 2
2, 3, i,–i
3-6 Fundamental Theorem of Algebra
Essential Question
• How do you write a polynomial equation of
least degree with given roots?
• Write the roots as factors using the
Fundamental Theorem of Algebra and
multiply. The number of roots will be the
degree of the polynomial.
Holt McDougal Algebra 2
3-6 Fundamental Theorem of Algebra
Q: Why did the polynomial plant wilt?
A: Its roots were imaginary.
Holt McDougal Algebra 2