ALGEBRA 5 EXPANDING AND SIMPLIFYING ALGEBRAIC EXPRESSIONS
Before continuing with this section you might like to revise the distributive property in Algebra 1 BEDMAS and the Distributive Property. An example is included here. Once again the italic form of 'x' represents the variable, and 'x' the times operator symbol.
+ x+ = +
+x- = -
- x - = +
Expand 1) 4(x + y)
4(x + y) = 4 x x + 4 x y
= 4x + 4y
Expand 2) 5(a - b)
5(a - b) = 5 x a - 5 x b (notice the '-' operator symbol)
= 5a - 5b
+ x+ = +
+x- = -
- x - = +
Expand 1) 4(x + y)
4(x + y) = 4 x x + 4 x y
= 4x + 4y
Expand 2) 5(a - b)
5(a - b) = 5 x a - 5 x b (notice the '-' operator symbol)
= 5a - 5b
Your turn.
Expand the following:
1) 3(x + y)
2) 12(x + z)
3) 6(m - n)
4) 9(a + b+ c)
5) 4(a + b - c)
6) -4(r + t) (negative numbers!)
7) -2(x + y)
8) -5(p - k)
Answers: 1) 3x+3y, 2) 12x+12z, 3) 6m-6n, 4) 9a+9b+9c, 5) 4a+4b-4c, 6) -4r-4t, 7) -2x-2y,
8) -5p+5k
Expand the following:
1) 3(x + y)
2) 12(x + z)
3) 6(m - n)
4) 9(a + b+ c)
5) 4(a + b - c)
6) -4(r + t) (negative numbers!)
7) -2(x + y)
8) -5(p - k)
Answers: 1) 3x+3y, 2) 12x+12z, 3) 6m-6n, 4) 9a+9b+9c, 5) 4a+4b-4c, 6) -4r-4t, 7) -2x-2y,
8) -5p+5k
numbers and variables together
"Expand (a) 5(x + 7)
5(x + 7) = 5 x x + 5 x 7 "The Distributive Property"
= 5x + 35
Expand (b) 4(3x - 1)
4(3x - 1) = 4 x 3x - 4 x 1
= 12x - 4
5(x + 7) = 5 x x + 5 x 7 "The Distributive Property"
= 5x + 35
Expand (b) 4(3x - 1)
4(3x - 1) = 4 x 3x - 4 x 1
= 12x - 4
Your turn.
Expand the following:
1) 2(x + 2)
2) 3(x + 5)
3) 4(x - 6)
4) 11(x - 4)
5) 6(x - 6)
6) -2(m + 3) (negative numbers!)
7) -8(r - 1)
8) -7(k + 9)
9) -5(p - 4)
10) -12(w + 12)
Answers: 1) 2x+4, 2) 3x+15, 3) 4x-24, 4) 11x-44, 5) 6x-36, 6) -2m-6, 7) -8r+1, 8) -7k-63,
9) -5p+20, 10) -12w-144
Expand the following:
1) 2(x + 2)
2) 3(x + 5)
3) 4(x - 6)
4) 11(x - 4)
5) 6(x - 6)
6) -2(m + 3) (negative numbers!)
7) -8(r - 1)
8) -7(k + 9)
9) -5(p - 4)
10) -12(w + 12)
Answers: 1) 2x+4, 2) 3x+15, 3) 4x-24, 4) 11x-44, 5) 6x-36, 6) -2m-6, 7) -8r+1, 8) -7k-63,
9) -5p+20, 10) -12w-144
More variables
Your turn.
Expand:
1) k(k + 1)
2) m(2m - 2)
3) w(w + 3)
4) 2n(n - 4)
5) 3j(2j + 1)
6) -5x(x + 8)
7) -8t(3t + 2)
8) -4p(6p - 3)
9) 6b(2b + a)
10 -7q(3q - p)
Expand:
1) k(k + 1)
2) m(2m - 2)
3) w(w + 3)
4) 2n(n - 4)
5) 3j(2j + 1)
6) -5x(x + 8)
7) -8t(3t + 2)
8) -4p(6p - 3)
9) 6b(2b + a)
10 -7q(3q - p)
Expanding and simplifying by collecting like terms
Examples: Expand and simplify by collecting like terms, 2(x + 3y) + 4(x + 6y).
Try doing the first expansion in your head (so you don't have to write 2 x x + 6 x y + 4 x x + 4 x 6y). You still can though if that helps!
2(x + 3y) + 4(x + 6y) = 2x + 6y + 4x + 24y and collecting like terms together
= 2x + 4x + 6y + 24y
=6x + 30y
Your turn
Expand and simplify
1) 4(a + b) + 5(a + b)
2) 3(2a + 3b) + 4(2a + 5b)
3) 2(3x + y) + 3(4x + 5y)
4) 3(4t + 3p) + 4(2t + 4p)
5) 5(k + 4m) + 3(k - 2m) tricky!
6) 8(3v + 2w) - 2(3v - 3w) very tricky!
Answers: 1) 4a+4b+5a+5b=9a+9b, 2) 6a+9b+8a+20b=14a+29b, 3) 6x+2y+12x+15y=18x+17y
4) 12t+9p+8t+16p=20t+25p, 5) 5k+20m+3k-6m=8k+14m, 6) 24v+16w-6v+6w=18v+22w
Try doing the first expansion in your head (so you don't have to write 2 x x + 6 x y + 4 x x + 4 x 6y). You still can though if that helps!
2(x + 3y) + 4(x + 6y) = 2x + 6y + 4x + 24y and collecting like terms together
= 2x + 4x + 6y + 24y
=6x + 30y
Your turn
Expand and simplify
1) 4(a + b) + 5(a + b)
2) 3(2a + 3b) + 4(2a + 5b)
3) 2(3x + y) + 3(4x + 5y)
4) 3(4t + 3p) + 4(2t + 4p)
5) 5(k + 4m) + 3(k - 2m) tricky!
6) 8(3v + 2w) - 2(3v - 3w) very tricky!
Answers: 1) 4a+4b+5a+5b=9a+9b, 2) 6a+9b+8a+20b=14a+29b, 3) 6x+2y+12x+15y=18x+17y
4) 12t+9p+8t+16p=20t+25p, 5) 5k+20m+3k-6m=8k+14m, 6) 24v+16w-6v+6w=18v+22w
expand and simplify double brackets
When you expand double brackets you get a whole lot of stuff happening that you can't see from just doing the math. That is because most functions can be represented visually in the form of a graph. When you multiply two x terms together you get a quadratic or x squared term such as:
The word 'quadratic' is derived from the Latin quadratus which means 'square' and takes the form of :
where a and b are coefficients and 'c' a constant. If we compare this to the equation above we would say the coefficient of x squared would be 1, the coefficient of x would be 5 and the constant would be 6. This really won't mean a lot of sense just yet. But when we look at the graphs for these functions later, something wonderful happens.
There are a number of methods used to simplify and expand two brackets. We will explore four methods and you can choose a method that works for you.
Example 1: Expand and simplify (x + 2)(x + 3) using the table or farmers field method
Example 1: Expand and simplify (x + 2)(x + 3) using the table or farmers field method
Your Turn
Expand and simplify
1) (x + 2)(x + 5)
2) (x + 6)(x + 4)
3) (x + 7)(x - 1)
4) (x - 3)(x - 4)
5) (x -1)(x + 5)
Expand and simplify
1) (x + 2)(x + 5)
2) (x + 6)(x + 4)
3) (x + 7)(x - 1)
4) (x - 3)(x - 4)
5) (x -1)(x + 5)
Using FOIL (first - outside- inside - last)
Expand and simplify (x + 2)(x + 3) using FOIL
FOIL stands for First, Outside, Inside and Last. It relates to the order that we multiply
the terms (variables and constants or letters and numbers) inside the brackets.
FOIL stands for First, Outside, Inside and Last. It relates to the order that we multiply
the terms (variables and constants or letters and numbers) inside the brackets.
repeat the same activity using foil below
Expand and simplify
1) (x + 2)(x + 5)
2) (x + 6)(x + 4)
3) (x + 7)(x - 1)
4) (x - 3)(x - 4)
5) (x -1)(x + 5)
1) (x + 2)(x + 5)
2) (x + 6)(x + 4)
3) (x + 7)(x - 1)
4) (x - 3)(x - 4)
5) (x -1)(x + 5)