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Year 1 Pure Maths
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- Pure
- Institution
- PEARSON (PEARSON)
Document with revision notes, explanations and examples for the whole of year 1 (year 12) pure maths.
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Pure: P1 P21. Algebraic expressions
1. 1. Index laws
● a ❑m × a❑n=a ❑m +n
● a ❑m ÷ a❑n=a❑m−n
● ( a ❑m ) ❑n=a ❑mn
● ( ab ) ❑m=a ❑m b ❑m
1. 2. Expanding brackets
To find the product of two expressions you must multiply each term in one expression by each
term in the other expressions.
1. 3. Factorising
It is the opposite of expanding brackets.
2
A quadratic expression has the form ax ❑ +bx+ c where a and b are real numbers and a≠0.
To factorise a quadratic expression:
1. Find two factors of ac that add up to b
2. Re-write the b term as a sum of the two factors
3. Factorise each pair of terms
4. Take out the common factor
Example:
For expressions in the form x ❑2− y ❑2=(x + y )( x− y )
,1. 4. Negative and fractional indices
Laws:
1
● a ❑m =m√ a
n
● a ❑m =m√ a❑n
−m 1
● a❑ = m
a❑
● a ❑0 =1
1. 5. Surds
Surds are numbers that can not be simplified to remove a root. It is an irrational number.
Laws:
● √❑
● √❑
● √❑
To simplify:
1. You must divide it into 2 factors, one of which is a square number
2. Write it as √ ❑
3. Do the square root of the square number
Example:
√❑
= √❑
=3 √ ❑
1. 6. Rationalising the denominator
If a fraction has a surd as the denominator, it is useful to rearrange it so that the denominator
is a rational number.
Rules:
If in the form… Do...
1
● , multiply top and bottom by √ ❑
√❑
1
● , multiply top and bottom by a−√ ❑
a+ √❑
1
● , multiply top and bottom by a+ √❑
a− √ ❑
2. Quadratics
2. 1. Solving quadratics
A quadratic expression can be written in the form ax ❑2 +bx+ c where a and b are real numbers
and a≠0.
,They can have one repeated root (solution), two different roots or no roots.
To solve by factorising:
1. Write in the form ax ❑2 +bx+ c=0
2. Factorise the left side
3. Set each factor to equal to 0 and solve to find the value(s) of x.
To solve with the quadratic formula:
1. Write in the form ax ❑2 +bx+ c=0
2. Substitute into the quadratic formula
−b ± √ ❑
x=
❑
2. 2. Completing the square
Quadratic expressions can be rewritten by completing the square.
Formulae:
2
( b2 )❑ −( b2 )❑
● x ❑ + bx= x +
2 2
● ax ❑ +bx+ c=a ( x+ ) ❑ + ( c−
4a )
2
2 b 2 b❑
2a
Example:
2. 3. Functions
A function is a mathematical relationship that maps each value of a set of inputs to a single
output. The notation f(x) is used to represent a function of x.
- The set of possible inputs for a function is called the domain
- The set of possible outputs of a function is called the range.
- The roots of a function are the values of x for which f(x) = 0.
To solve for x:
1. Substitute the input for every x in the given formula for the function.
, 2. 4. Quadratic graphs
When f(x) = ax ❑2 +bx+ c , the graph y=f(x) has a curved shape called a parabola.
- When a is positive, the parabola will have a U shape.
- When a is negative the parabola will have a ¿❑ ¿❑❑ shape.
- The graph crosses the y-axis when x=0. The y-coordinate is equal to c.
- The graph crosses the x-axis when y=0. The x-coordinates are the roots of the function
f(x).
- Quadratic graphs have a turning point. This can be minimum or maximum. Since a
parabola is symmetrical, the turning point is halfway between the roots.
To find the coordinates of the turning point:
1. Complete the square.
2
2. Once you have f(x) = a ( x + p ) ❑ + q, you know that the turning point will be (-p,q).
2. 5. The discriminant
If you square any real number, the result is greater than or equal to 0. This means that if y is
negative, then √❑ is not a real number.
The discriminant expression:
2
b ❑ −4 ac
- Ifb ❑2−4 ac > 0, then f(x) has 2 different roots.
- Ifb ❑2−4 ac = 0, then f(x) has 1 repeated root.
- Ifb ❑2−4 ac < 0, then f(x) has no real roots.
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