7. Algebraic Methods

Question 1

Dividing polynomials by a binomial

Answer 1

1. Write the equation under the bus stop and the binomial to the left
2. Divide the highest power of x in the polynomial by the x in the binomial and write above the value of x to that power in the original.
3. Multiply that by each value in the binomial and write below the corresponding power of x in the polynomial
4. Subtract those from the polynomial, bringing down any values in the polynomial that you have subtracted zero from
5. Repeat until you reach 0 or a whole number remainder
6. Write as polynomial/binomial = answer + remainder/binomial

Question 2

(b-a)/(a-b)

Answer 2

-1

Question 3

If f(p) = 0 in the function f(x)

Answer 3

f(x-p) is a factor of f(x)

Question 4

Fully factorising a cubic

Answer 4

1. Substitute values that are factors of the number on its own until f(that value) = 0
2. In that case (x- that value) is a factor
3. Use polynomial division to find a quadratic
4. Factorise the quadratic and write down those factors with the first factor found using f(x) = 0

Question 5

How to find an unknown coefficient from a factor

Answer 5

Find what x makes the bracket equal 0 (p)
Substitute in to the equation
Find what value of the coefficient makes f(p) = 0

Question 6

Proving coordinates form a parallelogram/rhombus

Answer 6

Prove that there are 2 sets of parallel sides and they aren’t perpendicular
Also prove that each length is the same for a rhombus

Question 7

Proof by exhaustion

Answer 7

1. Write down every possible value
2. Test each and say why it is/isn’t true
3. Make a concluding statement

Question 8

Disproof by counter example

Answer 8

Find one example where the rule doesn’t hold and then make a concluding statement

Question 9

A ⇒ B

Answer 9

A implies B
If A is true B is true
If A is false B could be true or false
If B is true A is true or false
If B is false A is false
A is sufficient for B but not necessary
B is necessary for A but not sufficient

Question 10

A ⇔ B

Answer 10

A implies and is implied by B
A and B will be the same
A is necessary and sufficient for B
B is necessary and sufficient for B

Question 11

A <= B

Answer 11

A is implied by B
If B is true then A is true
If B is false then A is true or false
If A is true then B could be true or false
If A is false then B is false
B is sufficient for A but not necessary
A is necessary for B but not sufficient

Question 12

Sufficient

Answer 12

It being true makes the other true

Question 13

Necessary

Answer 13

The other can only be true if it is true