Everything covered in this lesson, in one place - useful for revision or printing.
\( (a+b)(a^2-ab+b^2) = a^3+b^3 \)
Algebra is the language of the whole Grade 10 year, so this lesson builds the tools you will use in every other topic.
By the end you will be able to:
• sort numbers into rational, irrational and non-real
• round off and place a surd between two integers
• expand products, including the special products and cubes
• factorise using common factors, grouping, difference of squares, trinomials and cubes
• simplify algebraic fractions, stating the restrictions
Use Next to move through the steps. Try each quick check before you open the worked answer.
\( \mathbb{N} \subset \mathbb{N}_0 \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \)
Every number you meet in Grade 10 belongs to a family.
• Rational \( (\mathbb{Q}) \): can be written as \( \dfrac{a}{b} \) with \( a, b \) integers and \( b \neq 0 \). Terminating and recurring decimals are rational.
• Irrational \( (\mathbb{Q}') \): real, but cannot be written as a fraction of integers, e.g. \( \pi \), \( \sqrt{2} \). Their decimals never end and never repeat.
• Non-real: the even root of a negative number, e.g. \( \sqrt{-4} \). No real number squared gives a negative.
Rational and irrational numbers together make up the real numbers \( \mathbb{R} \).
\( 0{,}75 = \dfrac{75}{100} = \dfrac{3}{4} \)
Show that \( 0{,}75 \) is rational.
Write the decimal over a power of 10, then simplify:
\( \sqrt{25} \qquad \sqrt{-7} \qquad \pi \)
Classify each number as rational, irrational or non-real.
Always simplify first — a root sign does not automatically mean irrational.
\( \text{rational} \;|\; \text{irrational} \;|\; \text{non-real} \)
Number system in a nutshell
• Rational = fraction of integers (terminating or recurring decimals)
• Irrational = real, but no fraction form (e.g. \( \pi \), \( \sqrt{2} \))
• Non-real = even root of a negative
• Simplify before you classify: \( \sqrt{25} = 5 \) is rational.
\( 34{,}4678 \approx 34{,}47 \)
Round \( 34{,}4678 \) to two decimal places.
Look at the digit just after the place you are rounding to:
\( 5 < \sqrt{33} < 6 \)
Between which two consecutive integers does \( \sqrt{33} \) lie?
Find the perfect squares on either side of 33:
\( 5 < \;?\; < 6 \)
Which of these is an irrational number between 5 and 6?
A. \( \sqrt[3]{28} \) B. \( \sqrt{20+8} \) C. \( \sqrt{20 \times 8} \) D. \( 2\pi \)
\( a^2 < n < b^2 \Rightarrow a < \sqrt{n} < b \)
Key ideas
• Rounding: look one digit past the required place; 5 or more rounds up.
• To place \( \sqrt{n} \): trap \( n \) between two perfect squares.
• For cube roots use perfect cubes: \( 27 < 28 < 64 \Rightarrow 3 < \sqrt[3]{28} < 4 \).
\( -3x^2y(5xy^2 + xy) \)
Multiply the term outside by every term inside. Multiply coefficients, add exponents of like bases.
\( (2x+3)(5-x) \)
Each term in the first bracket multiplies each term in the second (four products), then collect like terms.
\( (2x-1)(x^2-3x+1) \)
Multiply each term of the binomial by all three terms of the trinomial (six products), then collect like terms.
\( (2r-p)(3r^2-4rp+p^2) \)
Two variables work exactly the same way. Keep the letters in the same order in every term so like terms are easy to spot.
\( (a+b)^2 = a^2 + 2ab + b^2 \)
Products checklist
• Every term in one bracket times every term in the other
• Coefficients multiply, exponents of like bases add
• Watch the signs, then collect like terms
• \( (a-b)^2 = a^2 - 2ab + b^2 \), never \( a^2 - b^2 \)
\( (a+b)(a^2-ab+b^2) = a^3+b^3 \)
Two products collapse to just two terms because the middle terms cancel:
\( (xy^3-3)(x^2y^6+3xy^3+9) \)
Let \( a = xy^3 \) and \( b = 3 \). Check the trinomial:
\( a^2 = x^2y^6 \), \( ab = 3xy^3 \), \( b^2 = 9 \). It matches \( (a-b)(a^2+ab+b^2) \).
\( \text{Area of picture} = \;? \)
A picture is framed using four rectangular pieces of wood of width \( x \). Read the lengths from the diagram and find the area of the picture in simplest form.
The frame is made of four pieces of width \( x \). Each \( 7x+3 \) length covers the picture plus one frame width, and so does each \( 5x+2 \) length.\( a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) \)
Special products
• \( (a+b)(a-b) = a^2 - b^2 \)
• \( (a \pm b)^2 = a^2 \pm 2ab + b^2 \)
• \( (a+b)(a^2-ab+b^2) = a^3 + b^3 \) and \( (a-b)(a^2+ab+b^2) = a^3 - b^3 \)
• Word problems: read lengths carefully from the diagram before multiplying.
\( xy^2 + 3x^2y = xy(y + 3x) \)
Factorising is expanding in reverse. Always look for a highest common factor (HCF) first.
\( x^2 - 7x - 18 \)
For \( x^2 + bx + c \), find two numbers that multiply to \( c \) and add to \( b \).
\( 6x^2 + 7x - 20 \)
When the \( x^2 \) coefficient is not 1, try factor pairs of the first and last terms until the inner and outer products add to the middle term.
\( \text{HCF} \rightarrow a^2-b^2 \rightarrow \text{trinomial} \)
Order of attack
• 1. Take out the HCF
• 2. Two terms, both squares, minus sign: \( a^2 - b^2 = (a-b)(a+b) \)
• 3. Three terms: find the pair that multiplies to the last and adds to the middle
• Always check by expanding.
\( 2px + 3qx - 2py - 3qy \)
Four terms usually means grouping: pair the terms so each pair has a common factor, and the brackets left over match.
\( x^2y - 16 + 4y - 4x^2 \)
Rearrange so the pairs share a factor:
\( x^3 - y^9 \)
Write each term as a cube: \( x^3 = (x)^3 \) and \( y^9 = (y^3)^3 \). So \( a = x \), \( b = y^3 \).
\( a^3 - b^3 = (a-b)(a^2+ab+b^2) \)
More tools
• Four terms: group into pairs with a common bracket
• Cubes: first bracket copies the sign; trinomial has the opposite middle sign, last sign positive
• \( x^2 + 4 \) (sum of squares) and the cube trinomial do not factorise further
• Keep going until no bracket can be factorised again.
\( \dfrac{x^2 - 25}{25x - 125} \)
You may only cancel factors, never separate terms. So factorise the numerator and denominator first.
The denominator may never be zero, so state the restriction.
\( \dfrac{27x^3 - 8}{27x^2 + 18x + 12} \)
Numerator: difference of cubes with \( a = 3x \), \( b = 2 \). Denominator: common factor 3.
\( \dfrac{y^2-y-2}{y^2-4} \times \dfrac{y^2+2y}{y^2+y} \)
Factorise every numerator and denominator, then cancel common factors across the whole product.
\( \dfrac{x^2-1}{(x+2) + x(x+2)} \div \dfrac{x-1}{2x+4} \)
To divide, multiply by the reciprocal of the second fraction. Then factorise and cancel.
\( \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \)
Simplifying fractions
• Factorise every numerator and denominator fully
• Cancel factors only, never terms
• Divide = multiply by the reciprocal
• State restrictions: any value that makes a denominator (or a divisor) zero is excluded.
\( \text{expand} \;\rightleftarrows\; \text{factorise} \)
You have covered the whole Grade 10 algebraic expressions topic.
• Rational, irrational and non-real numbers; rounding; surds between integers
• Products of binomials and trinomials, squares, sum and difference of cubes
• Factorising: HCF, difference of squares, trinomials, grouping, cubes
• Algebraic fractions: factorise, cancel factors, multiply and divide, state restrictions
Expanding and factorising are opposite processes — check every factorisation by expanding it again.