Polynomials

Question 1

Consider the polynomial P(x) = 3x^4 - 2x^2 - 6x + 21
State:
a) The degree of P(x) 
b) The leading coefficient of P(x)
c) The leading term of P(x)
d) The constant term of P(x)

Answer 1

a) The degree of P(x) = 4
b) The leading coefficient of P(x) = 3
c) The leading term of P(x) = 3x^4
d) The constant term of P(x) = 21

Question 2

What is (α + ß) in a quadratic equation?

Answer 2

[(-b)/a] -> negative b divided by a

Question 3

What is (α x ß) in a quadratic equation?

Answer 3

(α x ß) = c/a or c divided by a

Question 4

Give (α + ß) the value of 1/2
Give (α x ß) the value of 1/4
Find the value of (α - ß)(ß - α)

Answer 4

(α - ß)(ß - α) = (αß + αß) - ß^2 - α^2 
(αß + αß) = (1/4 + 1/4)
(1/2) - ß^2 - α^2 = x
1/2 - (α + ß)^2 + 2αß = x
1/2 - (1/2)^2 + 2(1/4) =x
1/2 - 1/4 + 2/4 = x
2/4 - 1/4 + 2/4 = x
3/4 = x 
3/4 = (α - ß)(ß - α)

Question 5

What is (α + ß + γ) in a cubic equation?

Answer 5

(-b/a) or negative b divided by a

Question 6

What is (αß + αγ + ßγ) in a cubic equation?

Answer 6

c/a or c divided by a

Question 7

What is (αßγ) in a cubic equation?

Answer 7

(-d/a) or negative d divided by a

Question 8

What is the remainder theorem?

Answer 8

The remainder theorem is the process of substituting the value that the polynomial is being divided by to find the remainder that the division will leave. E.g. 3x^2 - 2x + 6 divided by x-2 can be done in the remainder theorem by substituting 2 in; 3(2)^2 -2(2) + 6 = 14. Therefore the remainder of this division is 14.

Question 9

What is a division identity?

Answer 9

A division identity in polynomials is the result of polynomial division written in the form: D(x) Q(x) + R(x).
This means that [(The Divisor x The Quotient) + The Remainder] is the division identity.

Question 10

How can you find the remainder of an equation if you know the polynomial and the divisor (P(x) & D(x))?

Answer 10

Substitute the value of the divisor into the polynomial. E.g. if D(x) = x-2, then substitute 2 into P(x). This answer is the remainder. This is the remainder theorem.

Question 11

What is the factor theorem?

Answer 11

The factor theorem is the process where you substitute values that are the factor of the product of the first coefficient and the last coefficient. After finding these factors, substitute each of them into P(x) to find what number needs to be put in to make the equation = 0. When you find this value, you know that (x - n) is a factor of P(x). [Let n be the value you substitute in].

Question 12

Find the value of k given that (x + 3) is a factor of x^3 + 4x^2 +kx - 12

Answer 12

Substitute -3 into each x position. This will leave you with -3 +k(-3) = 0
Move the -3 across so that -3k =3
3/-3 = -1
Therefore, the value of k = -1
You can check this by seeing if (-3)^3 + 4(-3)^2 + -1(-3) - 12 = 0 [It does]

Question 13

A combination is a grouping of objects is where order is _______

Answer 13

irrelevant.

Question 14

What are some key-words that show that a question is a combination, rather than a permutation?

Answer 14

Group, Committee, Team, etc

Question 15

What are the roots of x^3 + 5x^2 + 8x + 4 = 0

Answer 15

x^3+5x^2+8x+4=0
(x+1)(x+2)^2=0
x = -1, x = -2

Question 16

What is factorial notation?

Answer 16

Factorial notation uses an exclamation mark (!) as a short way to write the product of consecutive positive integers

n! = (n − 1)(n − 2)× ⋯×3×2×1

Question 17

Simplify 12!/7!

Answer 17

=12!/7!
=12x11x10x9x8x7!/7!
=12x11x10x9x8

Question 18

A group of 8 girls and 7 boys decide to go to the park
a) How many ways can the boys sit together in a row
b) How many ways can the girls sit together in a row
c) How many different ways can the whole group sit together
in a row

Answer 18

a) 7! x 9!
b) 8! x 8!
c) 15!

Question 19

What is the multiplication principle?

Answer 19

If an outcome can happen in m different ways, and a second outcome can happen in n different ways, then the total number of ways in which two outcomes can happen together is m x n.

Question 20

How many 6 letter words can be formed in which the second, fourth and sixth letters are vowels and the other three letters are consonants?

Answer 20

21×5×21×5×21×5 = 1 157 625

Question 21

What is a permutation?

Answer 21

A permutation or ordered set is an arrangement of objects chosen from a certain set, without repetition. Order is important!

Question 22

What is a “complementary” situation?

Answer 22

In some situations, it may be easier to deal with the complementary event first – i.e. you count the unacceptable orderings, then you subtract that from the total number of orderings. Sometimes the question uses the word ‘not’ to prompt you, however often they don’t!

Question 23

How do you calculate arrangements around a circle?

Answer 23

The number of ways of arranging n different objects in a circle, regarding clockwise and anticlockwise arrangements as different can be shown to be:

n!/n = (n-1)

This is because in a circular arrangement, there is no start or
finish, so to count the arrangements, one object needs to be
fixed as the ‘starting object’

Question 24

How many ways can four people: Samuel, Mikaela, Oliver and Meira be arranged in a circle?

Answer 24

3! = 6 was

Question 25

How do you calculate the arrangements of objects when some are repeated?

Answer 25

The number of ways of arranging n objects in a row when p of the objects are identical and q of the objects are identical (but different to the others) and so on, is not n!, but rather:

n!/p!q!

Question 26

In how many ways can the 13 letters of the word Woolloomooloo be arranged in a line:

Answer 26

13!/8!x3! = 926640 ways

Question 27

What is a combination?

Answer 27

A combination (or selection) is an unordered permutation of all or part of a set of objects. Unlike permutations, for combinations, the order is not important!

Question 28

What is the pigeonhole principle?

Answer 28

The pigeonhole principle is a way of understanding how a number of items can be placed into a number of containers (ie. if you have five pigeons and four pigeonholes, then
the only way that you can place the remaining pigeon is to put it in with one of the pigeons already in a pigeon hole)

Question 29

How many terms are there in the expansion of (1+x)^n

Answer 29

The expansion of 1 + x has 2 terms, (1 + x)^2 has 3 terms, (1 + x)^3 has 4 terms … so in general, the expansion of (1 + x)^n has n+1 terms