Polynomial Review

The Basics

100

What is a polynomial?

A polynomial is a monomial or a sum or difference of monomials.

200

What is the degree of the following polynomial

3x^(2) - 4 +8x^(4)

The degree of

3x^(2) - 4 +8x^(4)

 is 4.

300

Subtract the following polynomials: 

(34 + 8x^3 - 9x^2) - (3x^3 + 10x^2 - 4x - 4)

(34 + 8x^3 - 9x^2) - (3x^3 + 10x^2 - 4x - 4) = 5x^3 - 19x^2 + 4x + 38

400

If h(x) =

x^2 - 6x + 3

, find h(-2)

If h(x) =

x^2 - 6x + 3

, 

h(-2) = (-2)^2 - 6(-2) + 3 = 19

500

Use a graphing calculator to find the zeros of 

f(x) = 2x^2 + 5x - 4

The zeros of

f(x) = 2x^2 + 5x + 3 

are -1 and -3/2

Dividing Polynomials

100

If the remainder after synthetic division is NOT zero, is (x - a) a factor of P(x)?

(x - a) is only a factor of P(x) if the remainder after synthetic division is zero. So if the remainder is NOT zero, (x - a) is NOT a factor.

200

If (x + 1/2) is our divisor, what do we put in the little box in order to perform synthetic division?

If (x + 1/2) is our divisor, we set (x - 1/2) equal to 0 to find what we put in the little box - in this case, it would be 1/2.

300

Divide the following polynomials: 

(2y^3 - y^2 + 25) divide (y - 3)

2y^2 + 5y + 15 + (70/(y-3))

400

Is (x - 4) a factor of

x^3 - 4x^2 + 3x - 5

 ?

Since

(x^3 - 4x^2 + 3x - 5) / (x - 4)

 does not have a remainder of 0, (x - 4) is not a factor of 

x^3 - 4x^2 + 3x - 5

500

Write the simplest polynomial with roots at x = -2, 3, and 1. Write your answer in standard form.

The simplest polynomial with roots at x = -2, 3, and 1 is 

x^3 - 2x^2 - 5x + 6

Factoring Polynomials

100

Factor the following polynomial: 

x^2 - 17x - 38

(x + 2) (x - 19)

200

Factor the following polynomial: 

4d^4 + 108d

4d (d^3 + 27) = 4d(d+3)(d^2 - 3d + 9)

300

Factor the following polynomial by grouping: 

4x^3 - 8x^2 - 9x + 18

(x - 2) (4x^2 - 9) = (x-2)(2x-3)(2x+3)

400

Factor the following polynomial 

(if you're having trouble, try factor by grouping!) 

2x^3 - x^2 -2x +1

(2x-1)(x-1)(x+1)

500

Whiteboard Daily Double!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Mr. Amalfitano will write down the question on the white board - make your wagers now.

And the answer is ...

Finding those Roots

100

What is the difference between a "zero" of a polynomial and a "root" of a polynomial?

There is no difference!

200

You can earn bonus points with this question: Write down all statements that are equivalent to the following sentence: "a is a root of the polynomial P(x)"

Possible answers: P(a) = 0 a is an x-intercept of the graph of P(x) (x - a) is a factor of P(x) When you divide P(x) by (x - a), the remainder is 0 a is a zero of P(x)

300

Is (x + 2) a factor of P(x) =

-2x^3 + 7x^2 - 3x - 9

 ?

(x + 2) is NOT a factor of

P(x) = -2x^3 + 7x^2 - 3x - 9

, because the remainder after synthetic division is NOT zero.

400

Find the roots of 

f(x) = x^2 + 3x - 5

The roots of

f(x) = x^2 + 3x - 5

 are 

x = (-3 +- sqrt(29))/2

500

Find all the roots of 

P(x) = x^4 - x^3 + 7x^2 - 9x - 18

The roots of

P(x) = x^4 - x^3 + 7x^2 - 9x - 18

 are: -1, 2, 3i, and -3i