1. Demonstrate Basic Algebra Skills
US20659
Demonstrate Basic Algebra Skills

2. LIKE TERMS LIKE terms: 2x, 3x, 31x UNLIKE terms: 2x, 3 4ab, 7ab
- LIKE terms are those with exactly the same letter, or combination of letters and powers
LIKE terms:
2x, 3x, 31x
UNLIKE terms:
2x, 3
4ab, 7ab
5x, 6x2
2ab, 2ac
e.g. Circle the LIKE terms in the following groups:
a) 3a 5b 6a 2c
b) 2xy 4x xy 3z yx
While letters should be in order, terms are still LIKE if they are not.

3. ADDING/SUBTRACTING TERMS
- We ALWAYS aim to simplify expressions from expanded to compact form
- Only LIKE terms can be added or subtracted
- When adding/subtracting just deal with the numbers in front of the letters
e.g. Simplify these expanded expressions into compact form:
a) a + a + a
= ( )a
b) 5x + 6x + 2x
= ( )x
= 3a
= 13x
c) 3p + 7q + 2p + 5q
= (3 + 2)p
( )q
d) 4a + 3a2 + 7a + a2 1
= 5p + 12q
= (4 + 7)a
( )a2
= 11a + 4a2
- For expressions involving both addition and subtraction take note of signs
e.g. Simplify the following expressions:
a) 4x + 2y – 3x
= (4 – 3)x
+ 2y
b) 3a – 4b – 6a + 9b
= x + 2y
= (3 - 6)a
( )b
= -3a + 5b
c) 3x2 - 9x + 6x2 + 8x - 5
= (3 + 6)x2
( )x
- 5
= 9x2 - x - 5
If the number left in front of a letter is 1, it can be left out
d) 4ab2 +2a2b – 5ab2 + 3ab
= -ab2 + 2a2b + 3ab

4. POWER RULES a) p10 × p2 = p(10 + 2) b) a3 × a2 × a = a(3 + 2 + 1) 1
1. Multiplication
- Does x2 × x3 = x × x × x × x × x ?
YES
- Therefore x2 × x3 = x5
- How do you get = 5 ? +
- When multiplying index (power) expressions with the same letter, ADD the powers.
No number = 1 i.e. p = 1p1
e.g. Simplify
a) p10 × p2
= p(10 + 2)
b) a3 × a2 × a
= a( ) 1
= p12
= a6
- Remember to multiply any numbers in front of the variables first
e.g. Simplify
a) 2x3 × 3x4
= 2 × 3
x(3 + 4)
b) 2a2 × 3a × 5a4 1
= 6 x7
= 2 × 3 × 5
a( )
= 30 a7

5. 2. Division
- Does 6 = 1 ? 6
YES
- Therefore x = 1 x
- Does x5 = x × x × x × x × x ?
x x × x × x
YES
= x × x × 1 × 1 × 1
- Therefore x5 = x2 x3
- How do you get = 2 ? -
- When dividing index (power) expressions with the same letter, SUBTRACT the powers.
e.g. Simplify
a) p5 ÷ p
= p(5 - 1)
b) x7 x4
= x(7 - 4) 1
= p4
= x3
- Remember to divide any numbers in front of the variables first
e.g. Simplify
a) 12x5 ÷ 6x4
= 12 ÷ 6
x(5 - 4)
b) a7
15a2
÷ 5
= 1 5
a(7 - 2)
= 2 x
= 1 5 a5
or a5 5
If the power remaining is 1, it can be left out of the answer

6. a) (c4)6 = c(4 × 6) b) (a3)3 = a(3 × 3) = c24 = a9 a) (3d2)3 = 33
3. Powers of powers
- Does (x2)3 = x2 × x2 × x2 ?
YES
- Does x2 × x2 × x2 = x6 ?
YES
- Therefore (x2)3 = x6
- How do you get = 6 ? ×
- When taking a power of an index expression to a power, MULTIPLY the powers
e.g. Simplify
a) (c4)6
= c(4 × 6)
b) (a3)3
= a(3 × 3)
= c24
= a9
- If there is a number in front, it must be raised to the power, not multiplied
e.g. Simplify
a) (3d2)3
= 33
× d(2 × 3)
b) (2a3)4
= 24
× a(3 × 4)
= 27 d6
= 16
a12
- If there is more than one term in the brackets, raise all to the power
e.g. Simplify
b) (4b2c5)2
= 42
b(2 × 2)
c(5 × 2)
a) (x3y z4)3
= x(3 × 3)
y(1 × 3)
z(4 × 3) 1
= x9 y3
z12
= 16 b4
c10

7. EXPANDING
- Does 6 × (3 + 5) = 6 × × 5 ?
YES
6 × 8 =
48 = 48
- The removal of the brackets is known as the distributive law and can also be applied to algebraic expressions
- When expanding, simply multiply each term inside the bracket by the term directly in front
e.g. Expand
a) 6(x + y)
= 6 × x
+ 6 × y
b) -4(x – y)
= -4 × x
- -4 × y
= 6x
+ 6y
= -4x
+ 4y
c) -4(x – 6)
= -4 × x
- -4 × 6
d) 7(3x – 2)
= 7 × 3x
- 7 × 2
= -4x
+ 24
= 21x
- 14
e) x(2x + 3y)
= x × 2x
+ x × 3y
f) -3x(2x – 5)
= -3x × 2x
- -3x × 5
= 2x2
+ 3xy
= -6x2
+ 15x
Don’t forget to watch for sign changes!

8. - If there is more than one set of brackets, expand them all then collect any like terms.
e.g. Expand and simplify
a) 2(4x + y) + 8(3x – 2y)
= 2 × 4x
+ 2 × y
+ 8 × 3x
- 8 × 2y
= 8x
+ 2y
+ 24x
- 16y
= 32x
- 14y
b) -3(2a – 3b) – 4(5a + b)
= -3 × 2a
- -3 × 3b
- 4 × 5a
+ -4 × 1b
= -6a
+ 9b
- 20a
- 4b
= -26a
+ 5b

9. EXPANDING TWO BRACKETS
- To expand two brackets, we must multiply each term in one bracket by each in the second
Remember integer laws when multiplying
e.g. Expand and simplify
a) (x + 5)(x + 2)
= x2
+ 2x
+ 5x
+ 10
b) (x - 3)(x + 4)
= x2
+ 4x
- 3x
- 12
= x2 + 7x + 10
= x2 + 1x – 12
To simplify, combine like terms
c) (x - 1)(x - 3)
= x2
- 3x
- 1x
+ 3
c) (2x + 1)(3x - 4)
= 6x2
- 8x
+ 3x
- 4
= x2 – 4x + 3
= 6x2 – 5x – 4

10. PERFECT SQUARES - When both brackets are exactly the same
To simplify, combine like terms
e.g. Expand and simplify
Watch sign change when multiplying
a) (x + 8)2
= (x + 8)(x + 8)
b) (x - 4)2
= (x – 4)(x – 4)
= x2
+ 8x
+ 8x
+ 64
= x2
- 4x
- 4x
+ 16
= x2 + 16x + 64
= x2 - 8x + 16
Write out brackets twice BEFORE expanding
c) (3x - 2)2
= (3x – 2)(3x – 2)
= 9x2
- 6x
- 6x
+ 4
= 9x2 – 12 x + 4

11. DIFFERENCE OF TWO SQUARES
- When both brackets are the same except for signs (i.e. – and +)
e.g. Expand and simplify
a) (x – 3)(x + 3)
= x2
+ 3x
- 3x
- 9
b) (x – 6)(x + 6)
= x2
+ 6x
- 6x
- 36
= x2 – 9
= x2 – 36
Like terms cancel each other out
c) (2x – 5)(2x + 5)
= 4x2
+ 10x
- 10x
- 25
= 4x2 – 25

12. FACTORISING - Factorising is the reverse of expanding - To factorise:
1) Look for a common factor to put outside the brackets
2) Inside brackets place numbers/letters needed to make up original terms
You should always check your answer by expanding it
e.g. Factorise
a) 2x + 2y
= 2( ) x
+ y
b) 2a + 4b – 6c
= 2( ) a
+ 2b
- 3c
- Always look for the highest common factor
e.g. Factorise
a) 6x - 15
= 3( ) 2x
- 5
b) 30a + 20
= 10( ) 3a
+ 2
- Sometimes a ‘1’ will need to be left in the brackets
e.g. Factorise
a) 6x + 3
= 3( ) 2x
+ 1
b) 20b - 10
= 10( ) 2b
- 1

13. a) 5a2 – 7a5 = a2( ) 5 - 7a3 b) 4b2 + 6b3 = 2b2( ) 2 + 3b
- Letters can also be common factors
e.g. Factorise
a) cd - ce
= c( ) d
- e
b) xyz + 2xy – 3yz
= y( ) xz
+ 2x
- 3z
c) 4ad – 8a
= 4 ( ) a d
- 2
- Powers greater than 1 can also be common factors
e.g. Factorise
a) 5a2 – 7a5
= a2( ) 5
- 7a3
b) 4b2 + 6b3
= 2b2( ) 2
+ 3b

14. FACTORISING QUADRATICS
The general equation for a quadratic is ax2 + bx + c
When a = 1
- You need to find two numbers that multiply to give c and add to give b
e.g. Factorise
To check answer, expand and see if you end up with the original question!
a) x2 + 11x + 24
= (x + 3)(x + 8)
1, 24
List pairs of numbers that multiply to give 24 (c)
Check which pair adds to give 11 (b)
Place numbers into brackets with x
2, 12
3, 8
4, 6
b) x2 + 7x + 6
= (x + 1)(x + 6)
1, 6
List pairs of numbers that multiply to give 6 (c)
Check which pair adds to give 7 (b)
Place numbers into brackets with x
2, 3

15. - Expressions can also contain negatives e.g. Factorise
a) x2 + x – 12
= (x - 3)(x + 4)
b) x2 – 6x – 16
= (x + 2)(x - 8) -
1, 12
1, 16 -
As the end number (c) is , one of the pair must negative.
As the end number (c) is , one of the pair must negative. -
2, 6
2, 8 - -
3, 4
4, 4 -
Check which pair now adds to give b
Make the biggest number of the pair the same sign as b
Check which pair now adds to give b
Make the biggest number of the pair the same sign as b
c) x2 – 9x + 20
= (x - 4)(x - 5)
d) x2 – 10x + 25
= (x - 5)(x - 5) -
1, 20 - -
1, 25 -
= (x - 5)2
As the end number (c) is , but b is – 9, both numbers must be negative -
2, 10 - -
5, 5 -
As the end number (c) is , but b is – 10, both numbers must be negative -
4, 5 -
Check which pair now adds to give b

16. SPECIAL CASES 1. No end number (c) e.g. Factorise a) x2 + 6x + 0
b) x2 – 10x
= x( )
x - 10
0, 6
= x(x + 6)
Add in a zero and factorise as per normal
OR: factorise by taking out a common factor
2. No x term (b) (difference of two squares)
e.g. Factorise
a) x2
- 25
+ 0x
= (x - 5)(x + 5)
b) x2 – 100
= (x )(x )
-5, 5
c) 9x2 – 121
= (3x )(3x )
Add in a zero x term and factorise
OR: factorise by using A2 – B2 = (A – B)(A + B)