Higher Order Polynomials

Question 1

What’s the general equation of a polynomial?

Answer 1

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀

Question 2

When do you use long division for polynomials?

Answer 2

Dividing P(x) by D(x), where the degree of D(x) is >= 2

Question 3

What are the formulas for long division of polynomials?

Answer 3

P(x) = D(x)Q(x) + R(x)
P(x) / D(x) = Q(x) + R(x) / D(x)

P(x) is the dividend, D(x) is the divisor, Q(x) is the quotient, R(x) is the remainder

Question 4

What are the steps for long division?

Answer 4

1. Divide leading term of P(x) by leading term of D(x)
2. Multiply divisor by result
3. Subtract that product from the dividend
4. Repeat until the remainder’s degree is less than the divisor’s degree
5. Practice

Question 5

What is the remainder theorem?

Answer 5

When P(x) is divided by (ax - b), the remainder is P(b/a)

Question 6

What is the factor theorem?

Answer 6

If (ax - b) is a factor of P(x), then P(b/a) = 0

Question 7

What is the rational zero theorem?

Answer 7

If (ax - b) is a rational zero of P(x), then b is a factor of the constant term and a is a factor of the leading coefficient.

Question 8

What are the steps to factor polynomials?

Answer 8

1. List all possible factors of the constant term and leading coefficient
2. Substitute b/a into P(x) until it is equal to 0
3. Divide P(x) by ax-b
4. Final answer in factored form

Question 9

State the properties of even order roots, odd order roots, and order of 1 roots

Answer 9

Even order roots: Bounce
Odd order roots: Wiggle
Order of 1 roots: Cross/cut

Question 10

State the end behaviour for an even and odd degree polynomial

Answer 10

Even degree polynomial: Both ends go up in the same direction
Odd order polynomial: Ends go in opposite directions

Question 11

State the 4 possible end behaviours given the degree of the polynomial and the sign of the leading coefficient.

Answer 11

Even degree + positive l.c: Both ends up
Even degree + negative l.c: Both ends down
Odd degree + positive l.c: Left falls, right rises
Odd degree + negative l.c: Left rises, right falls

Question 12

How do you solve polynomial inequalities?

Answer 12

Method 1: Graphing
Method 2: Factor table using critical points

Question 13

What is a critical point?

Answer 13

A point on the graph where the function’s behaviour changes, specifically related to local maximum and minimum values or changes in concavity

Question 14

What is the sum of roots for a polynomial of degree n?

Answer 14

-a(subscript n-1) / a(subscript n)

Question 15

What is the product of roots for a polynomial of degree n?

Answer 15

(-1)^n times a(subscript 0) / a(subscript n)

Question 16

When a polynomial has a degree of n, how many roots does it have?

Answer 16

It has n roots

Question 17

How do you use pascal’s triangle for binomial expansion?

Answer 17

For (a + b)^n, there are 2 key patterns
1. Powers of a decrease from n to 0, by 1 each time
2. Powers of b increase from 0 to n, by 1 each time