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

= (x + 1) (x2 – 4x – 2x + 8)

f (x) = (x + 1) (x – 2) ( x – 4) Write f (x) in factored form.

= (x + 1) (x2 – 6x + 8)Multiply.= x3 – 6x2 + 8x + x2 – 6x + 8

Multiply.

Combine like terms.

Combine like terms.

Use the three zeros and the factor theorem to write f(x) as a product of three factors.

SOLUTION

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros – 1, 2, 4.

x=−1, x+1=0, x=2, x−2=0, x=4, x−4=0

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 − 4f(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.

= x3 – 7x2 + 11x + 3 Combine liketerms.

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

= f(x) =(x – 3)[x – (3 – i)][x –(3 + i)]

= (x–3)[(x– 3)+i ][(x2 – 3) – i] Regroup terms. = (x–3)[(x – 3)2 –i2)]

= (x– 3)[(x – 3)+ i][(x –3) –i]Multiply.

Write f (x) in factored form.

SOLUTION

= (x – 3)[(x – 3)2 – i2]=(x –3)(x2 – 6x + 9)

Simplify.= (x–3)(x2 – 6x + 9)= x3–6x2 + 9x – 3x2 +18x – 27

Combine like terms.= x3 – 9x2 + 27x –27Multiply.

Descartes’ Rule of Signs

French mathematician Rene Descartes (1596-1650) 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 SignsDetermine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f (x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.

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

SOLUTION

Use Descartes’ Rule of SignsThe 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

The coefficients in f (x) have 1 sign changes, so f has 1 positive real zero(s).

SOLUTION

f (x) = 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.

f (– x) = (– x)3 + 2(– x) – 11

= – x3 – 2x – 11

10. g(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

SOLUTION f (x) = 2x4 – 8x3 + 6x2 – 3x + 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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