3.6 Theorems About Polynomials

3.6 Theorems about Polynomials

Take notes on this slide show in your notebooks.

​Example Question: How many zeros should the polynomial p(x) = x⁴-5x⁷+2x-1 have?

Answer: p(x) should have 7 solutions since the degree (highest exponent when the polynomial is in standard form) of p(x) is 7.

Multiple Choice

What degree is the following polynomial?

$$f\left(x\right)=2x-5x^3+3x^7-9x^2+14$$  

• 1

• 3

• 7

• 2

Multiple Choice

Based on the Fundamental Theorem of Algebra, how many real and complex roots exist for this polynomial: $$f\left(x\right)=2x-5x^3+3x^7-9x^2+14$$  

• 1

• 3

• 7

• 2

Multiple Choice

How many solutions does the equation

$$0=4x^2+2x-1$$  have?

• 0

• 1

• 2

• 3

Multiple Choice

How many zeros does the function

$$g\left(x\right)=x^4-8x-2$$  have?

• 3

• 1

• 0

• 4

Multiple Choice

how many zeros does the function

$$f\left(x\right)=x\left(x-2\right)\left(x+1\right)$$  have?

• 0

• 1

• 2

• 3

​Example Question: What is the lowest degree possible of a polynomial with roots x = -3i and x = 2+i.

Answer: Since all imaginary roots come in pairs, we know that x = 3i and x = 2-i are also roots of the polynomial. With that said, the lowest degree is 4.

Multiple Choice

f is a polynomial with real coefficients and

$$2-i$$  is a zero of f, what is another solution of f(x)?

• -2 - i

• -2 + i

• 2 + i

• 2 - i

Multiple Choice

g is a polynomial function with real coefficients and

$$-2i$$  is a zero of g. What is another zero?

• 1-2i

• 1+2i

• 2i

• -2i

Multiple Choice

f is a polynomial function with real coefficients. If

$$2-\sqrt{5}i$$  is a zero of f, what is another zero of f?

Jon & Jon's Mom

• $$2+\sqrt{5}i$$  

• $$2-\sqrt{5}i$$  

• $$-2-\sqrt{5}i$$  

• $$-2+\sqrt{5}i$$  

​Example Question: Determine the remainder when p(x) = x³-4x²-7x+10 is divided by (x-2).

Answer: Evaluate p(2). The answer is the remainder.

Multiple Choice

You Try!

What is the remainder if $$P\left(x\right)=x^4-3x^2-2x+5$$  is divided by $$x+2$$  ?

• 13

• 23

• -13

• -23

Multiple Choice

You Try!

If $$p\left(x\right)=2x^3-3x+5$$  , what is the remainder of p(x) $$\div$$  (x-5)?

• -230

• 0

• 40

• 240

FACTOR THEOREM

​Example Question: Is x+1 a factor of p(x) = x¹¹ - 4x - 3?

Answer:

If p(-1) = 0, then the remainder must be zero when p(x) is divided by (x+1), which would mean (x+1) is a factor. So first evaluate p(-1).

p(-1) = (-1)¹¹ - 4(-1) - 3

p(-1) = -1 + 4 - 3

p(-1) = 0

x+1 is a factor of p(x)!

FACTOR THEOREM

Multiple Choice

You Try!

Is $$x+1$$   a factor of $$x^{40}-1$$  ?

• Yes

• No

Multiple Choice

You Try!

Is $$x-3$$   a factor of $$x^5+16x-200$$  ?

• Yes, because when we evaluate the polynomial for x = 3, we get 0.

• Yes, because when we evaluate the polynomial for x = 3, we get 3.

• No, because when we evaluate the polynomial for x = 3, we get 91.

• No, because when we evaluate the polynomial for x = 3, we get -491.

Multiple Choice

Is $$x-3$$  a factor of $$3-7x+5x^2-x^3$$  ?

• Yes

• No

Multiple Choice

Is $$2x-1$$  a factor of $$2x^3+3x^2-8x+3$$  ?

• Yes

• No