Algebra

Question 1

Modulus properties

Answer 1

|a x b|=|a|x|b|
|a/b| = |a| / |b|

Note:
|a+b|≠|a|+|b|
y =|sin(x)|≠ sin|x|

Question 2

Division of polynomials

Answer 2

f(x) ≡ (x – a) * q(x) + R
dividend ≡ divisor x quotient + remainder

Where the degree of the remainder is less than the degree of the divisor.

Question 3

Steps for polynomial division

Answer 3

1. Lay out equation as long division
2. Ensure all powers of x are written including 0 values
3. To begin filling in the quotient or the answer multiply the divisor by a value that when subtracted by the dividend will cancel that power
4. Continue to do this till you are left with the remainder
5. To check remainder, if its degree is less than the degree of the divisor it is correct

Question 4

Remainder theorem

Answer 4

Set divisor equal to 0
Solve for the divisor
Sub answer for divisor into the dividend
Solve and then answer is the remainder

Question 5

Factor theorem

Answer 5

Use factors of the constant in the dividend as the trial inputs to find factors of the dividend

Question 6

Partial fractions Type 1

Answer 6

Denominator contains linear factors only

  ax + b                      A                 B ------------------       =   ---------    +   --------- (px + q)(rx+s)           (px + q)        (rx + s) 

ax + b = A(rx+s) + B(px + q)

Question 7

Partial fractions Type 2

Answer 7

Denominator contains repeated linear factors

ax² + bx + c A B C
—————— = —— + —— + ——(px + q)(rx+s)² (px+q) (rx+s) (rx+s)²

ax² + bx + c =
A(rx+s)² + B(px+q)(rx+s) + C(px + q)

Question 8

Partial fractions Type 3

Answer 8

Denominator contains a quadratic factor that cannot be factorised

ax² + bx + c A Bx + C
—————— = ——— + ———
(px + q)(rx²+s) (px + q) (rx² + s)

ax² + bx + c =
A(rx²+s) + (Bx + c)(px + q)

Question 9

Binomial Expansion

Answer 9

Given in formula sheet

Note:
(a + x)ⁿ
= aⁿ (1 + x/a)ⁿ
= aⁿ[ 1 + n(x/a) + (n(n-1) / 2!)(x/a)² + …]

Question 10

Binomial expansion for complex equations

Answer 10

Split into partial fractions
Binomially expand each part of the fraction
Add the expansions to obtain answer

Note:
For rational functions where the degree of the numerator is greater than or equal to the degree of the denominator (improper fractions), try express them in another way using long division to then expand.