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6-6 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 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
P(x) = (x – 0)(x –
P(x) = (x2 –
2
3
2
3
x)(x – 3)
2
P(x) = x3– 11
x
+ 2x
3
Holt Algebra 2
)(x – 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.
6-6 Fundamental Theorem of Algebra
Complex Conjugate Root Theorem:
If 𝑎 + 𝑏𝑖 is a root of a polynomial equation, then
𝑎 − 𝑏𝑖 is also a root.
Irrational Root Theorem:
If 𝑎 + 𝑏 𝑐 is a root of a polynomial equation, then
𝑎 − 𝑏 𝑐 is also a root.
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Example 3: Writing a Polynomial Function with
Complex Zeros
Write the simplest function with zeros 2i,
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: 2i, -2i,
,
, and 1. The polynomial
must have degree 5.
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Example 3 Continued
Step 2 Write the equation in factored form.
P(x) = [x + (2i)][x – (2i)](x +
)[(x –
)](x – 1)
Step 3 Multiply.
P(x) = (x2 – 4)(x2 – 3)(x – 1)
= (x4 – 7x2+ 12)(x – 1)
5
4
3
2
P(x) = x – x – 7x + 7x + 12x + 12
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Check It Out! Example 3
Write the simplest function with zeros 1 + 2i,
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: 1+2i, 1–2i, and 3. The polynomial must have
degree 3.
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
Check It Out! Example 3 Continued
Step 2 Write the equation in factored form.
P(x) = [ x – (1+2i)][x – (1-2i)](x - 3)
Step 3 Multiply.
P(x) = (x2 – x(1-2i) – x(1+2i) + (1+2i)(1-2i))(x – 3)
2
P(x) = (x – x + 2xi – x – 2xi + 1 – (2i)2)(x – 3)
P(x) = (x2 – 2x – 5)(x – 3)
P(x) = x3 – 5x2 – 11x + 15
Holt Algebra 2
6-6 Fundamental Theorem of Algebra
HW pg. 449
#’s 12, 13, 36, 39, 41, 43
Holt Algebra 2