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2.7 Apply the Fundamental
Theorem of Algebra day 2
How do you use zeros to write a polynomial
function?
What is Descartes’ rule of signs?
Use zeros to write a polynomial function
Write a polynomial function f of least degree
that has rational coefficients, a leading
coefficient of 1, and the given zeros – 1, 2, 4.
Use the three zeros and the factor theorem to
write f(x) as a product of three factors.
x=−1, x+1=0, x=2, x−2=0, x=4, x−4=0
SOLUTION
f (x) = (x + 1) (x – 2) ( x – 4)
Write f (x) in factored form.
= (x + 1) (x2 – 4x – 2x + 8)
= (x + 1) (x2 – 6x + 8)
= x3 – 6x2 + 8x + x2 – 6x + 8
Multiply.
Combine like terms.
Multiply.
Combine like terms.
Complex Conjugates theorem
If f is a polynomial function with real
coefficients, and a + bi is an imaginary zero of
f, then a −bi is also a zero of f.
Imaginary numbers always travel in pairs!
Using Zeros to Write Polynomial Functions
Write a polynomial function f of least degree that
has real coefficients, a leading coefficient of 1,
and 2 and 1 + i as zeros.
x = 2, x = 1 + i, AND x = 1 − i.
Complex conjugates always travel in pairs.
f(x) = (x − 2)[x − (1 + i )][x − (1 − i )]
f(x) = (x − 2)[(x − 1) − i ][(x − 1) + i ]
f(x) = (x − 2)[(x − 1)2 − i2 ]
f(x) = (x − 2)[(x2 − 2x + 1 −(−1)]
f(x) = (x − 2)[x2 − 2x + 2]
f(x) = x3 − 2x2 +2x − 2x2 +4x − 4
f(x) = x3 − 4x2 +6x − 4
Irrational Conjugates Theorem
Use zeros to write a polynomial function
SOLUTION
Write f (x) in
factored form.
Regroup terms.
= (x – 3)[(x – 2)2 – 5]
Multiply.
= (x – 3)[(x2 – 4x + 4) – 5]
Expand binomial.
= (x – 3)(x2 – 4x – 1)
Simplify.
= x3 – 4x2 – x – 3x2 + 12x + 3
Multiply.
Combine like
terms.
= x3 – 7x2 + 11x + 3
8. 3, 3 – i
Because the coefficients are rational and 3 –i is a zero,
3 + i must also be a zero by the complex conjugates
theorem. Use the three zeros and the factor theorem
to write f(x) as a product of three factors
SOLUTION
Write f (x) in
= f(x) =(x – 3)[x – (3 – i)][x –(3 + i)] factored form.
= (x–3)[(x– 3)+i ][(x2 – 3) – i]
Regroup terms.
= (x–3)[(x – 3)2 –i2)]
Multiply.
= (x– 3)[(x – 3)+ i][(x –3) –i]
= (x – 3)[(x – 3)2 – i2]=(x –3)(x2 – 6x + 9)
= (x–3)(x2 – 6x + 9)
Simplify.
= x3–6x2 + 9x – 3x2 +18x – 27
Multiply.
= x3 – 9x2 + 27x –27
Combine like terms.
Descartes’ Rule of Signs
French mathematician
Rene Descartes (15961650) found a
relationship between
the coefficients of a
polynomial functions
and the number of
positive and negative
zeros of the function.
Descartes’ Rule of Signs
Use Descartes’ Rule of Signs
Determine the possible numbers of positive real
zeros, negative real zeros, and imaginary zeros
for
f (x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.
SOLUTION
f (x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.
The coefficients in f (x) have 3 sign changes,
so f has 3 or 1 positive real zero(s).
f (– x) = (– x)6 – 2(– x)5 + 3(– x)4 – 10(– x)3 – 6(– x)2 – 8(– x) – 8
= x6 + 2x5 + 3x4 + 10x3 – 6x2 + 8x – 8
Use Descartes’ Rule of Signs
The coefficients in f (– x) have 3 sign changes,
so f has 3 or 1 negative real zero(s) .
The possible numbers of zeros for f are
summarized in the table below.
Determine the possible numbers of positive
real zeros, negative real zeros, and
imaginary zeros for the function.
9.
f (x) = x3 + 2x – 11
SOLUTION
f (x) = x3 + 2x – 11
The coefficients in f (x) have 1 sign changes,
so f has 1 positive real zero(s).
f (– x) = (– x)3 + 2(– x) – 11
= – x3 – 2x – 11
The coefficients in f (– x) have no sign
changes.
The possible numbers of zeros for f are
summarized in the table below.
10.
g(x) = 2x4 – 8x3 + 6x2 – 3x + 1
SOLUTION f (x) = 2x4 – 8x3 + 6x2 – 3x + 1
The coefficients in f (x) have 4 sign changes, so f
has 4 positive real zero(s).
f (– x) = 2(– x)4 – 8(– x)3 + 6(– x)2 + 1
= 2x4 + 8x + 6x2 + 1
The coefficients in f (– x) have no sign changes.
The possible numbers of zeros for f are summarized in
the table below.
• How do you use zeros to write a polynomial
function?
If x = #, it becomes a factor (x ± #). Multiply
factors together to find the equation.
• What is Descartes’ rule of signs?
The number of positive real zeros of f is equal
to the number of changes in sign of the
coefficients of f(x) or is less than this by an
even number.
The number of negative real zeros of f is equal
to the number of changes in sign of the
coefficients of f(−x) or is less than this by an
even number.
Assignment is p. 141, 20-28
even,34-40 even
Show your work
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