1 - Factor & Remainder Theorem

How do you multiply fractions?

(a / b) * (c / d) = ac / bd

How do you add fractions?

(a / b) + (c / b) = (a + c) / b

How do you divide fractions?

(a / b) / (c / d) = (a / b) * (d / c) = ad / bc

Show that (x - 1) is a factor of x^3 + 3x^2 - 4x + 1

(x - p) = (x - 1) so p = 1 f(1) = 1^3 + 3(1)^2 - 4(1) + 1 = 0 f(1) = 0 So (x - 1) is a factor

Simplify 7 / 2 and highlight the quotient and remainder

7 / 2 = 3 + (1 / 2) The quotient is 3 The remainder is the 1 on top of the / 2

Simplify (x^4 + x^3 + 2x) / x^3 and highlight the quotient and remainder

= x + 1 + (2x / x^3) x + 1 is the quotient 2x is the remainder

What is a root?

A value that can substitute for x so that f(x) = 0 The solutions to f(x) are also the roots of f(x)

Find the remainder when dividing (x^3 + x^2 + 2x + 2) by (x + 1)

Divisor = x + 1 So a = -1 Remainder = f(-1) f(-1) is where you substitute every x in f(x) for f(-1)

What is factor theorem?

If f(p) = 0 then (x - p) is a factor of f(x)

How else can you use the previous 4 rules?

In simplifying algebraic fractions

How do you subtract fractions?

Same as adding fractions

How do you simplify fully (3x^5 - 4x^2) / (-3x)

Split into (3x^5 / -3x) - (4x^2 / -3x) = - x^4 + (4x / 3)

If you are given a question where f(x) has some coefficients replaced with a, b etc. How do you find the values of these coefficients?

The question should give you some divisors and their remainders You evaluate f(a) = R for each of these and simplify them This gives you 2 equations concerning a and b These are simultaneous equations You can find the values of a and b in this way

How do you divide a function by another function when they can't be simplified? E.g (x^3 + 3x^2 - 4x + 1) / (x - 1)

Use algebraic long division

How do you fully factorise a cubic?

Use factor theorem and trial and error to find the first factor: Try f(1), f(-1), f(2), f(-2) etc until you find a factor Then use long division to divide f(x) by the factor to get a quadratic Factorise the quadratic as you normally would The factor we got from the factor theorem and the 2 factors from factorising the quadratic make up the 3 factors of the cubic: f(x)

(x + 1) is a factor of 4x^4 - 3x^2 + a Find a

We know that f(-1) = 0 because (x + 1) is a factor So we know that: 4(-1)^4 - 3(-1)^2 + a = 0 Simplify this and you get: a = -1 So f(x) = 4x^4 - 3x^2 - 1

When is remainder theorem used?

When you are given a function and a divisor and are asked to find the remainder but not the quotient

What is the remainder theorem?

When you divide f(x) by (x - a), the remainder is f(a)

If you are asked to divide f(x) by 2x + 1, what do you do?

f(a) = root of 2x + 1 2x + 1 = 0 2x = -1 x = -0.5 So f(a) = f(-0.5)