Expressions, Structure & Units — Print

Before you manipulate an expression, read it. Its structure tells you what it means and what you are allowed to do.

What the practice questions 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.

Read this out loud: 3x2 + 5x - 8.

Before you do anything to an expression, you have to see how it is built — the way you see where the words are in a sentence.

an expression is built out of parts, like a sentence 3x² + 5x − 8 term term term the + and − signs are the spaces between the words

The + and - signs are the spaces. They cut the expression into three terms you can handle one at a time.

And the minus belongs to what follows it. The third term is -8, not 8 — that single habit prevents a great many sign errors later.

Inside a term, everything is multiplied. 3x2 is 3 times x times x.

That difference is why 3x3 = x but 3x + 63 = x + 2 and never x + 6. You may cancel a factor; you may never cancel a single term.

The parts of an algebraic expression reading an expression is half of algebra 3 x 2 + 5 x - 8 coefficient variable exponent term term constant term 3 terms degree 2 one variable + and - signs cut an expression into terms; the sign in front belongs to the term after it so the third term is -8, not 8

Terms and factors

+ and - signs cut an expression into terms. Multiplication and division join things into factors.

3x2 + 5x - 8 has three terms. 3x2 itself has factors 3, x and x.

The sign in front belongs to the term after it. The third term is -8, not 8.

Why it matters

You may cancel a common factor from a fraction. You may never cancel a single term. 3x3 is x; 3x + 63 is x + 2, not x + 6.

Seeing structure

x4 - 16 looks like a quartic, but it is also (x2)2 - 42 — a difference of two squares.

(a + b)2 - 9 has the same shape, with a + b playing the part of the single thing being squared.

Looking for a familiar shape inside an unfamiliar expression is one of the most useful habits in algebra.

Cover it up

Put your finger over the complicated part and ask what shape is left. If x4 - 16 becomes □2 - 16, you know it factors as (□ - 4)(□ + 4).

Units and quantities

A formula's units tell you whether you have used it correctly. Distance divided by time gives speed; distance times time gives nothing meaningful.

When converting, multiply by a fraction equal to 1: 60 min1 hr. Arrange it so the unwanted unit cancels.

Choose a scale that fits the data. Measuring a country's population in units of one person is technically right and practically useless.

Cancel the units too

45 mi1 hr × 1 hr60 min = 0.75 mi/min. If the units do not cancel down to what you want, the fraction is upside down.

Evaluating and rewriting

To evaluate, substitute with brackets and follow the order of operations.

To rewrite, use the distributive property and collect like terms. Every step must keep the expression equivalent.

Check equivalence by trying a value. If x = 2 gives different answers, you made a mistake somewhere.

Test with a number

Not sure if 2(x + 3) equals 2x + 6? Try x = 5: 2(8) = 16 and 10 + 6 = 16. It is not proof, but it catches nearly every real error.

Worked examples

Naming the parts

In 7x3 - 2x + 11, name the number of terms, the degree, the coefficient of x, and the constant.

  1. Three terms: 7x3, -2x, +11 The + and - signs mark the boundaries.
  2. Degree 3, from the highest exponent. That is x3.
  3. The coefficient of x is -2. The sign belongs to the coefficient.
  4. The constant is 11. The term with no variable.

3 terms, degree 3, coefficient -2, constant 11

Using structure

Factor x4 - 81 by seeing its structure.

  1. Notice x4 = (x2)2 and 81 = 92. A difference of two squares in disguise.
  2. (x2)2 - 92 = (x2 - 9)(x2 + 9) Apply a2 - b2 = (a - b)(a + b).
  3. And x2 - 9 is itself a difference of squares. Keep going while you can.
  4. = (x - 3)(x + 3)(x2 + 9) x2 + 9 does not factor over the real numbers.

(x - 3)(x + 3)(x2 + 9)

Unit conversion

A tap runs at 18 litres per minute. Convert to litres per hour.

  1. Multiply by 60 min1 hr Minutes must cancel, so minutes goes on top.
  2. 18 L1 min × 60 min1 hr The min units cancel diagonally.
  3. = 1080 L per hour Sensible: an hour is longer, so the number is bigger.

1080 L/hr

Words to know

Term
A piece separated by + or -. In 3x2 + 5x - 8 there are three terms.
Coefficient
The number multiplying a variable.
Factor
Something being multiplied. In 5(x + 2) the factors are 5 and (x + 2).
Degree
The largest exponent on the variable.
Equivalent expressions
Two expressions that give the same value for every input.

Mistakes to watch for

Cancelling a term out of a fraction. In 3x + 63 the 3 on the bottom divides the whole top, so both terms must be divided: x + 2. Striking out one 3 answers a different question.

Dropping the minus from a coefficient. In 9 - 5x the coefficient of x is -5. The sign in front is part of the number, and forgetting it flips the behaviour of the whole term.

Setting up a conversion so the unwanted unit does not cancel. Multiply by 60 min1 hr and the minutes must appear top and bottom somewhere. If they do not cancel, the fraction is upside down.

Confusing the degree with the number of terms. 7x3 - 2x + 11 has three terms and degree three by coincidence; x5 + 1 has two terms and degree five. Degree is the largest exponent, nothing else.

For most of history algebra was written out in words. "The thing plus five equals twelve" — every problem, a paragraph.

François Viète began using letters for unknowns in 1591, and Descartes fixed the convention we still use: letters from the end of the alphabet (x, y, z) for unknowns, from the start (a, b, c) for knowns.

x became the default unknown partly because early French printers had plenty of spare x blocks — y and z were needed for ordinary French words.

Problem 1

How many terms are in 4x2 - 7x + 2?

Worked answer

3

4x2, -7x, +2

Three terms

Problem 2

What is the coefficient of x in 9 - 5x?

Worked answer

-5

The term is -5x

So the coefficient is -5

Problem 3

What is the degree of 3x4 - x2 + 8?

Worked answer

4

Exponents are 4, 2, 0

The degree is 4

Problem 4

Evaluate 3a - 2b when a = 5 and b = 4.

Worked answer

7

3(5) - 2(4)

= 15 - 8 = 7

Problem 5

Evaluate x2 - 3x when x = -2.

Worked answer

10

(-2)2 = 4

-3(-2) = 6

4 + 6 = 10

Problem 6

Simplify 6x + 93.

Worked answer

2x + 3

6x3 = 2x

93 = 3

2x + 3

Problem 7

Simplify 5(2x - 3) - 4x.

Worked answer

6x - 15

10x - 15 - 4x

= 6x - 15

Problem 8

Factor x2 - 49 using structure.

Worked answer

(x - 7)(x + 7)

x2 - 72

= (x - 7)(x + 7)

Problem 9

A car uses 9 litres per 100 km. How many litres for 250 km?

Worked answer

22.5

9 L100 km × 250 km

= 22.5 L

Problem 10

Convert 72 km/h to metres per second.

Worked answer

20

72 × 1000 = 72000 m per hour

72000 ÷ 3600 = 20

20 m/s

Problem 11

Which is equivalent to 4x + 84?

  1. x + 2
  2. x + 8
  3. 4x + 2
  4. x
Worked answer

x + 2

4x4 + 84

= x + 2

Cancelling only the first 4 would be wrong

Problem 12

Evaluate 2a + ba - b when a = 5 and b = 3.

Worked answer

6.5

Top: 10 + 3 = 13

Bottom: 5 - 3 = 2

13 ÷ 2 = 6.5

Problem 13

Factor x4 - 16 completely.

Worked answer

(x - 2)(x + 2)(x^2 + 4)

(x2)2 - 42 = (x2 - 4)(x2 + 4)

x2 - 4 = (x-2)(x+2)

x2 + 4 does not factor

Problem 14

A recipe needs 34 cup per serving. How many cups for 14 servings?

Worked answer

10.5

34 × 14 = 424

= 10.5 cups

Problem 15

Write 16x2 - 24x as a product of two factors.

Worked answer

8x(2x - 3)

GCF of 16x2 and 24x is 8x

8x(2x - 3)

Problem 16

If P = 2l + 2w with P = 46 and l = 14, find w.

Worked answer

9

46 = 28 + 2w

2w = 18

w = 9

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