Polynomials
Posted in Maths, Core 1A polynomial is an expression which consists of terms with positive integer powers, e.g. x3 + 2x + 1.
Expanding brackets
To expanding brackets you need to make sure every term is multiplied by every other term. When dealing with two brackets, such as (x - 1)(x + 2) this is relatively simple but when dealing with 3, 4 or even more brackets it can become quite messy. To make the process of expanding 3 or more brackets simpler you should work in stages as demonstrated below.
Question: Expand the following: (x + 2)(x + 1)(x + 4)2
Worked answer: Firstly it's best to rewrite the (x + 4)2 part of the expression as (x + 4)(x + 4) leaving us with (x + 2)(x + 1)(x + 4)(x + 4). From here we work in stages from right to left, first of all you expand the two end brackets (x + 4)(x + 4) to get:
(x + 2)(x + 1)[x2 + 8x + 16]
Next you work on this expression and the next bracket to the left, (x + 1). Firstly multiply each term in the square brackets by x to give: x3 + 8x2 + 16x. Then multiply each term in the square brackets by +1 to give x2 + 8x + 16. This now replaces both the square brackets and the (x + 1) bracket to leave you with:
(x + 2)[x3 + 8x2 + 16x + x2 + 8x + 16]
You can now collect like terms, that is adding up all the cubed terms, squared terms and constants (all terms with the same power) to give:
(x + 2)[x3 + 9x2 + 24x + 16]
Lastly you repeat the same process as before by multiplying each term of the squared brackets both x and, in this instance, +2, the two terms of the next bracket. This gives:
x4 + 9x3 + 24x2 + 16x + 2x3 + 18x2 + 48x + 32
Collecting like terms leaves you with the final expanded expression:
x4 + 11x3 + 42x2 + 64x + 32
Factorisation
Factorisation is the opposite of expansion, it deals with putting an expression such as x3 + x2 - 14x - 24 into brackets.
The remainder theorem
The remainder theorem can be helpful when dividing polynomials. The remainder theorem says that:
For a polynomial, f(x), f(a) is the remainder when f(x) is divided by (x - a)
Question: What is the remainder when f(x) = x2 + 2x + 1 is divided by (x - 2)?
Worked answer: Using the remainder theorem the remainder is simply f(a), in this case f(2) = 22 + 2 x 2 + 1 = 9. The remainder is 9
The factor theorem
The Factor theorem, as the name suggests, is very helpful in factorising polynomials. It says that:
For a polynomial, f(x), if f(a) = 0 then (x - a) is a factor of f(x)
What does this mean in simple terms? If when you put any whole number, a, into a polynomial, f(x), if the result is 0 then (x - a) goes into that polynomial a whole number of times, it is factor. This is used to factorise long polynomials:
Question: Factorise f(x) = x3 + x2 - 14x - 24
Worked answer: Factorisation can be a bit of trial an error; In this case we are looking for at most 3 brackets (as 3 is the highest power) which when multiplied together give the above expression. The constant of the expression, -24, is made from multiplying the 3 numbers in each of the brackets so these numbers could be 2, 3 and 4. The only way to know is to test them using the factor theorem. If f(2) is 0 then (x - 2) is a factor:
23 + 22 - 14 x 2 - 24 = -40
(x - 2) is clearly not one of the brackets. However before dismissing it you should also test for the negative, i.e. if f(-2) is 0 then (x + 2) is a factor:
(-2)3 + (-2)2 - 14 x (-2) - 24 = 0
Hurrah! This means (x + 2) is one of other three brackets. We now move onto our second number, 3/-3 and see if (x - 3) or (x + 3) is a factor:
33 + 32 - 14 x 3 - 24 = -30
(-3)3 + (-3)2 - 14 x (-3) - 24 = 0
So again the negative test shows that f(-3) = 0 and so (x + 3) is also a factor. You could go on to substitute 4 and -4 into f(x) to make sure however this, in this case, isn't required. -24 is needed from the 3 numbers and so far you have +3 and +2, hence the last number must be -4 in order for them to multiply together to get -24. Therefore the original polynomial, f(x), factorises as:
(x + 2)(x + 3)(x - 4)