Expressions, Structure & Units — Challenge
Before you manipulate an expression, read it. Its structure tells you what it means and what you are allowed to do.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Factor — A number that divides evenly into another.
- GCF — Greatest common factor — the largest number dividing both.
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Difference of squares — a2 - b2 = (a - b)(a + b).
Problem 1
Simplify 3x+2y-x+4y.
- 2x+6y
- 4x+6y
- 2x+2y
- 8xy
- 2x-6y
Worked answer
2x+6y
Only like terms can be combined, so handle the x terms and the y terms separately.
3x-x = 2x and 2y+4y = 6y.
The result is 2x+6y. The x and y terms cannot be merged further, which rules out 8xy.
Problem 2
Expand (x+3)(x+2).
- x2+6x+5
- x2+5x+6
- x2+6
- x2+5x+5
- 2x+5
Worked answer
x2+5x+6
Multiply each term of the first bracket by each term of the second.
x· x = x2, x· 2 = 2x, 3· x = 3x and 3· 2 = 6.
Collecting the middle terms gives x2+5x+6: the numbers add to make the x coefficient and multiply to make the constant.
Problem 3
Simplify 2(3x-4)+5.
- 6x-8
- 6x+1
- 6x-3
- 11x-4
- 6x-13
Worked answer
6x-3
Expand the bracket first: 2· 3x = 6x and 2·(-4) = -8.
So the expression is 6x-8+5.
Combining the constants gives 6x-3.
Problem 4
What is the value of 2x2-3 when x=-2?
- -11
- 5
- 8
- 13
- 29
Worked answer
5
Square before multiplying: (-2)2 = 4, which is positive.
Then 2· 4 = 8, and 8-3 = 5.
Treating 2x2 as (2x)2 would give 16-3 = 13, a different expression.
Problem 5
Simplify x· x· x.
- 3x
- x3
- x2
- 3x3
- x
Worked answer
x3
Three copies of x multiplied together is x to the power 3.
So the answer is x3. Three copies added together would be 3x, which is different.
Problem 6
Factorise x2-9.
- (x-9)(x+1)
- (x-3)(x+3)
- (x-3)2
- (x+3)2
- x(x-9)
Worked answer
(x-3)(x+3)
This is a difference of two squares: x2-9 = x2-32.
A difference of squares factorises as (x-3)(x+3).
Expanding back gives x2+3x-3x-9 = x2-9, since the middle terms cancel. That cancelling is why (x-3)2 is wrong.
Problem 7
Simplify 5xx, where x is not zero.
- 0
- 1
- 5
- 5x
- x
Worked answer
5
The x on the top and the x on the bottom divide out, leaving the 5 behind.
So the expression equals 5 for every non-zero x.
The condition x≠ 0 matters, because the original expression has no value when x=0.
Problem 8
Simplify 4a-7a.
- 3a
- -3a
- 11a
- -11a
- -3
Worked answer
-3a
Both terms are multiples of a, so subtract the coefficients.
4-7 = -3, so the answer is -3a.
The a does not vanish, which rules out -3.