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Maths Core 2

Simplifying Fractions

...something nice and easy to start the topic...

Whenever you are given a fraction with a single term appearing in the denominator, which appears in the numerator, that term can be cancelled (obviously).

For example:

= 3x³ - 2x² + 5. Easy.

It gets a little harder with a term like (x-2) on the bottom though, but that will usually require you factorising the top.

E.g.:

= = (x-2)

Hint: You know you've got it right because the denometer in this kind of equation is usually one of numerator terms.

Slightly trickier is when you have to factorise both top and bottom first, but it's still easy enough. :)

For example:

= =

All pretty easy, and should remind you of GCSE.

Long Division

Now it gets interesting. Ever wondered how to factorise a x³ equation? With great difficulty... But here's the first of a few methods that will allow you to factorise any polynomial equation. (polynomial means 'many terms'. x + x³ + x² + x + 1 for example.)

Remember WAAY back in primary school how they taught that horrible long division method?

Well, it's back with a vengeance. A reminder with numbers to start with:

The answer you get above is called the quotient. If you have a remainder, this is just called the remainder.

Now, that method may be extremely long-winded, but it is the method you need to know for solving algebraic divisions. Take the following example:

The first step is to divide x3 by x, to get x2 above the line. Then below, you times the x2 by x - 4.

Next, take the x3 - 4x2 from x3 - 5x2. If you're doing it right, the first two should cancel. Remember your brackets and - sign. Because we have -4x2, we end up adding 4x2. Write the answer below:

Then bring down -5x. Divide the -x2 by x to get -x and write this above. Times -x by x-4 and write the answer below.

Take (-x2 + 4x) from (-x2 + 5x). Be careful with the minuses! Here, if you're doing it right, the x2 should cancel. Write the answer below and then bring down -4.

Divide the x by x, and write 1 above. Finally times 1 by x-4 and write below. This will cancel and leave 0. You have reached the answer.

When x3 - 5x2 - 5x - 4 is divided by x - 4 the quotient is x2 - x + 1.

Unfortunately, not all questions they might give you in the exam will be nice and divide exactly. In that case, you'll get a remainder, and you'll need to be even more careful with your working out.

The questions might also miss out an x term. For example, in: (x3 - 2) (x - 3), there's no x2 or x term. You need to write these into the division with 0xn and thus get:

From there, you just use the same method as before to solve:

and from that you get a quotient of x2 + 3x + 9 and a remainder of 25.

From this we know that x3 - 2 is the same as: (x2 + 3x + 9)(x - 3) + 25.

Factor Theorem

The nice thing about maths is that when you know the hard method, you can find the easier one. For algebraic division, we use something called factor theorem and remainder theorem.

Factor theorem:

For any polynomial f(x), if f(a) = 0, then (x - a) is a factor of f(x).
If f() = 0, then (ax - b) is a factor.

This gives us the ability to find the factors of polynomials or prove them.

Show that (x - 2) is a factor of the polynomial (x3 - 2x2 + x - 2).

Using factor theorem we know that when f(a) = 0, (x - a) is a factor. Therefore, we can say that f(2) = 0, if it's a factor.

Now put 2 in for the values of x:

f(2) = 23 - 2x22 + 2 - 2

Therefore:

f(2) = 8 - 8 = 0

Because it equals 0, we know (x - 2) is a factor.

You can use trial and improvement to find the factors:

Factorise 2x3 + 3x2 - 32x + 15 completely.

Let: f(x) = 2x3 + 3x2 - 32x + 15

f(1) = 2 + 3 - 32 + 15 0 (therefore, (x - 1) is not a factor).

f(-1) = -2 + 3 + 32 + 15 0 (therefore, (x + 1) is not a factor).

We can rule out 2 and -2 because they aren't factors of 15.

f(3) = 54 + 27 - 96 + 15 = 0

Therefore (x - 3) is a factor!

To factorise completely, you then divide (2x3 + 3x2 - 32x + 15) by (x - 3). It should come out with no remainder - a little proof to check your factor is right:

From this you can say: (x - 3)(2x2 + 9x - 5) = 2x3 + 3x2 - 32x + 15

Factorise the quadratic to get an answer of: (x - 3)(2x - 1)(x + 5).

Remainder Theorem

Finally, the last part of simplifying equations is remainder theorem. After learning to use factor theorem, this is easy.

If f(x) is divided by (x - a), then f(a) gives the remainder.
If f(x) is divided by (ax - b), f() gives the remainder.

What's the remainder when f(x) = x3 +4 is divided by (x - 2)?

So f(x) = x3 + 4

=> f(2) = 23 + 4

23 + 4 = 12, so the remainder is 12.

Using this principle, the exam might ask a slightly harder version:

f(x) = ax3 + bx2 - 13x + 6. When f(x) is divided by (x + 1) the remainder is 18 and (2x - 1) is a factor of f(x). Find the values of a and b.

First, it's a case of writing in what you know:

For (x + 1) we get:

f(-1) = (a x -13) + (b x -12) - (13 x -1) + 6 = 18

18 = -a + b + 13 + 6

-a + b = -1

For a factor of (2x - 1) we get:

f() = (a x 3) + (b x 2) - (13 x ) + 6 = 0

0 = a + b - + 6

a + 2b = 4

From here, it's a case of simultanious equations to solve:

-a + b = -1 ... re-arrange ... b = a - 1

a + 2b = 4

...Sub in: a + 2(a - 1) = 4

=> a + 2a - 2 = 4

3a = 6

a = 2

Sub a into one of the equations to get b:

b = 2 - 1

b = 1

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Sine and Cosine Rules

More stuff that should be vaguely familiar from GCSE...

To fine a side using the Sine Rule, you use the following formula:

For an angle, it's simply the inverse:

(Capital letters for the angles, lower case for the sides ALWAYS.)

To illustrate, take the following shape:

Before to do anything, it's important to label on your A, B, C, a, b, and c, so that you know what you're working with. You can label any angle A B or C, but just make sure the side opposite it has the corresponding lower case letter.

Finding is easy here, because it's just a matter of knowing that angles in a triangle add up to 180°. Therefore: 180° - 96° = 84°.

To find y, you need two parts of the sine rule:

Put numbers in and manipulate and solve the equation to get:

A little reminder of bearings

Bearings are those nasty random things to do with angles they threw in at GCSE. They return at A-level, and it's even more important you understand them. Just bear with them, okay?

  1. Whenever drawing a bearing, always start from the vertical northline and measure clockwise.
  2. If you're finding a bearing of A from B, the from is the starting point. e.g.:

Special cases of Sine might have two solutions. (The exam question will usually hint towards this if it says find the solutions.) Here's what it might looks like:

For this to happen, the given angle (30° in the example) must be smaller than the two angles we're finding.

  1. Use Sine rule to work out the smallest angle (49.1° in the example).
  2. Do 180° - 49.1° = 130.9 to work out the other angle.

Cosine Rule

The formula for finding a side is: a² = b² + c² - 2bc.CosA.

For an angle, it's:

If the angle is acute, CosA is negative.
If the angle is obtuse, CosA is positive.

Find b in the following triangle:

b2 = a2 + c2 - 2ac.CosB

b2 = 12 + 22 - 2x1x2xCos60
b2 = 3
b = 3

Find the smallest angle in the following triangle:

Note: The smallest angle is always opposite the smallest side. An easier way that finding every angle!

Remember, to remove Cos from C, just do the inverse of Cos.

Area of a circle

This simple formula lets you work out the area of any circle. Pretty cool:

Area = ab SinC

Area = ac SinB

Area = 0.5x2.5x5xSin100

Area = 4.31 cm2


Exponentials and Logarithms

What is an exponential function? Well, you might recall the graph from GCSE...

y = ax

a is a constant and could be any number, while x is a power.

For example, the graphs of y = 2x and y = -2x:

y=2x

y=-2x

As you can see, the minus is a reflection in the y-axis. (not surprising, if you consider Core 1 Tranformations).

While the lines get very close, they never cross the x or y-axis. Therefore, the x and y-axis are asymptotes.

Logarithms

Another strange maths word... Look at the following:

25 = 32

This is written out in standard form. Another way of writing it would be in log form:

5 = log232

Both mean exactly the same. However, because your magic calculator can work out log10 numbers, logarithms become very useful. Before we get ahead of ourselves, let's explain what each bit means...

log7343 = 3

  • The little 7 is called the base number. When changed to index form, this is the original number that has a power applied to it.
  • The number after the equals sign is the power.
  • The number after the base is the result of 73.

In general:

xa= b logxb = a

Three important rules for manipulating logs:

1. logax + logay = logaxy

2. logax - logay = loga()

3. logaxn = nlogax

IMPORTANT NOTE: These only work when the bases are the same.

Proofs:

  1. Let logax = p and logay = q:

    Swap to index form: ap = x and aq = y

    xy = ap x aq = a(p + q)

    Convert back to logs: logaxy = p + q

    Sub in p and q again and you get:

    logaxy = logax + logay

  2. Let logax = p and logay = q:

    Swap to index form: ap = x and aq = y xy = ap aq = a(p - q)

    Convert back to logs: loga() = p - q

    Sub in p and q again and you get:

    loga() = logax - logay

  3. Let logax = p

    Swap to index form: x = ap

    xn = (ap)n = apn

    Convert back to logs: logaxn = pn

Sub in p again and you get:

logaxn = nlogax

Examples:

Simplify: loga8 - loga4 + loga5

(Remove with rule 3) = loga8 - loga4 + loga5

= loga2 - loga4 + loga5

(Use rule 2 for subtract) = loga + loga5

= loga0.5 + loga5

(use rule 1 for addition = loga0.5 x 5

= loga2.5

So, it's all very well and good messing around with logs, but how can we USE them? It's quite simple really with the calculator. Take the following example:

2x = 15.8

How can you figure out what x is? Well, you could use trial and error, or you could convert it all to base 10 form (which we just write as log):

log 2x = log15.8

Next, use the laws we know already:

x log 2 = log15.8

x =

Finally, stick that into your calculator and out pops the answer:

x = 3.98

Solve: 6(2x+1) = 3-x

log 6(2x+1) = log 3-x

(2x + 1)(log 6) = -x log 3

2x log 6 + log 6 = -x log 3

2x log 6 + x log 3 = -log 6

x(2 log 6 + log 3) = -log 6

x =

 


 

Trigonometric Functions

The Special Trig Angles:

You MUST learn the following table:

  Sin Cos Tan
30° (rad) ½

60° (rad)

½ √3
45° (rad) 1

 

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