Transcript Holt Algebra 2

Fundamental
Theorem
of
6-6
6-6 Fundamental Theorem of Algebra
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
Algebra
Holt
Algebra
22
6-6 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 Algebra 2
6-6 Fundamental Theorem of Algebra
Example 1: Writing Polynomial Functions
Write the simplest polynomial with roots –1, 2
,
3
and 4.
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.
2
8
P(x) = x3 – 11
x
–
2x
+
3
3
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Check It Out! Example 1a
Write the simplest polynomial function with
the given zeros.
–2, 2, 4
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.
P(x) = x3– 4x2– 4x + 16
Holt Algebra 2
6-6 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
x – r is a factor of P(x).
Multiply the first two binomials.
Multiply the trinomial by the
binomial.
2
P(x) = x3– 11
x
+ 2x
3
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Notice that the degree of the function in Example
1 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. Polynomials
functions, like quadratic functions, may have
complex zeros that are not real numbers.
Holt Algebra 2
6-6 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.
p = –36, and q = 1.
Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.
Find the real roots at
or near –1 and 4.
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Example 2 Continued
Test the possible real roots.
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Example 2 Continued
The polynomial factors into (x + 1)(x – 4)(x2 + 9) = 0.
The solutions are 4, –1, 3i, –3i.
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Check It Out! Example 2
Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all
roots.
Find the real roots at
or near –5 and 1.
Holt Algebra 2
6-6 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 Algebra 2
6-6 Fundamental Theorem of Algebra
Check It Out! Example 2 Continued
2
The polynomial factors into (x + 5)(x – 1)(x + 4) = 0.
The solutions are –5, 1, –2i, +2i).
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
HOMEWORKPage 449 # 1-6, 11-19
Holt Algebra 2