Mathematics C1 - Algebra and Functions 1

Key notes on part 1 of Core mathematics 1 (algebra and functions)

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  • Created on: 04-05-11 16:56
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Pure Mathematics Algebra and Functions 1
Indices
1) am x an = am+n Or 52x 53 = 55 = 3125
2) am÷ an = am-n Or 54÷52 = 52 = 25
3
3) (am)n = amn Or (52) = 56 = 15625
4) a-n = a
1
n Consider 7-3 = 1
73
1
= 343
1 1 3
5) a n = the nth root of a Consider 7 3 = 7
1 2
7 2 = 7 = 7
3 3
16 2 = (16) = 43 = 64
m 2
6) a n =
the nth root of a raised to the power m .
7) a0 = Consider 73÷73 = 73-3 = 70 = 1
1for any value of a which is not zero
Surds
Numbers like 2, 3, 11 are
called surds.
A surd is an irrational number which means that it cannot be written exactly as a decimal number or in
the form a b where a and b are integers.
The following rules apply: ab = a x b
a a
b = b

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Polynomial expressions
A polynomial is and algebraic expression which is the sum of a number of terms. They have four
terms. The highest power of x is 3.…read more

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Quadratic Functions
The graph of y = ax2 + bx + c
.
i:a>0 ii : a < 0
All quadratic functions are symmetrical about a vertical axis.
The x-coordinates of the point(s) where the curve crosses the x-axis are determined by solving
ax2 + bx + c = 0
The y-coordinates of the point where the curve crosses the y-axis is c. This is called the y-intercept.…read more

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Completing the square
Consider the expression x2 + 6x + 11.
We can write x2 + 6x = (x + 3)2 - 9 and so we say that
x2 + 6x = (x + 3)2 - 9 + 11
= (x + 3)2 + 2 (NOTE: the `3' is obtained from halving the `6', which is the coefficient of
x)
Simultaneous equations
There are two methods for solving two linear simultaneous equations.…read more

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And 3x + 7y = 44 (2)
From (1) we see that x = 13 - 2y
Substitute in (2), 3(13 - 2y) + 7y = 44
39 + y = 44
y = 5
Substitute this value in (1), x = 13 - 2(5)
x = 3
We also need to be able to solve simultaneous equations in which one of the equations is linear and
one is not linear.…read more

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Sketch the graph of y = x2 + x - 6
We are looking below the x-axis so we want one region, and hence one inequality.
So the solution is - 3 < x < 2
Example 2
1
Solve y = x2 + 1
2 x - 120
We factorise to give (x + 1.5)(x - 1) 0
The critical values are -1.…read more

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