Exploring Algebraic Identities

The One Thing

Algebraic identities are equalities that remain true for every permissible value of their variables. They provide systematic methods for expanding, simplifying, evaluating, and factorising expressions by applying the distributive property and recognising standard patterns.

Definitions and Results

• Algebraic expression: A combination of variables, constants, and mathematical operations, such as $3x+5$ or $a^2-b^2$.
• Term: A part of an expression separated by addition or subtraction signs, such as $4x$ and $7$ in $4x+7$.
• Coefficient: The numerical factor multiplying a variable, such as $6$ in $6y$.
• Identity: An equality that is true for every value of the variables for which both sides are defined.
• Expansion: The process of removing brackets by multiplying each term according to the distributive property.
• Factorisation: The process of writing an expression as a product of two or more factors.
• Distributive property:

$$a(b+c)=ab+ac$$ and $$a(b-c)=ab-ac.$$ It allows multiplication across a bracket.

• Common factor: A factor shared by every term of an expression, which can be taken outside brackets.
• Visual area model: A diagram representing products as areas of rectangles or squares. It explains why algebraic identities are valid by relating a total area to the sum of component areas.
• Polynomial: An algebraic expression made from variables, coefficients, and non-negative integer powers, such as

$$2x^2+3x-5.$$

Important identities include:

• Product of two binomials:

$$(x+a)(x+b)=x^2+(a+b)x+ab.$$

• Square of a sum:

$$(a+b{)}^2=a^2+2ab+b^2.$$

• Square of a difference:

$$(a-b{)}^2=a^2-2ab+b^2.$$

• Product of a sum and a difference:

$$(a+b)(a-b)=a^2-b^2.$$

• Identity involving three terms:

$$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca.$$

• Common-factor factorisation:

$$ax+ay=a(x+y).$$

• Difference of two squares:

$$a^2-b^2=(a+b)(a-b).$$

• Perfect-square trinomials:

$$a^2+2ab+b^2=(a+b{)}^2$$ and $$a^2-2ab+b^2=(a-b{)}^2.$$

An identity may be checked numerically for selected values, but only an algebraic proof establishes that it is true for all permissible values. Expansion and factorisation are inverse processes: expansion removes factors, whereas factorisation creates them.

Worked Methods

1. Expanding a single bracket

Apply the distributive property by multiplying the factor outside the bracket by every term inside it.

$$a(b+c)=ab+ac$$

For a subtraction:

$$a(b-c)=ab-ac.$$

The signs must be retained when multiplying negative terms.

2. Expanding two binomials with a common first term

For

$$(x+a)(x+b),$$

multiply each term in the first bracket by each term in the second:

$$(x+a)(x+b)=x^2+bx+ax+ab.$$

Collect the like terms:

$$x^2+bx+ax+ab=x^2+(a+b)x+ab.$$

Therefore,

$$(x+a)(x+b)=x^2+(a+b)x+ab.$$

This is also the identity for multiplying binomials with a common first term.

3. Expanding the square of a sum

Write the square as a product:

$$(a+b{)}^2=(a+b)(a+b).$$

Expand:

$$(a+b)(a+b)=a^2+ab+ab+b^2.$$

Combine the two $ab$ terms:

$$(a+b{)}^2=a^2+2ab+b^2.$$

An area model gives the same result. Divide a square of side $a+b$ into four regions with areas $a^2$, $ab$, $ab$, and $b^2$. The total area is therefore

$$a^2+ab+ab+b^2=a^2+2ab+b^2.$$

4. Expanding the square of a difference

Write the square as a product:

$$(a-b{)}^2=(a-b)(a-b).$$

Multiply term by term:

$$(a-b)(a-b)=a^2-ab-ab+b^2.$$

Hence,

$$(a-b{)}^2=a^2-2ab+b^2.$$

The negative middle term results from the two products involving one $a$ and one $-b$.

5. Expanding a product of a sum and a difference

Use the distributive property:

$$(a+b)(a-b)=a^2-ab+ab-b^2.$$

The middle terms cancel:

$$(a+b)(a-b)=a^2-b^2.$$

This gives the difference-of-two-squares identity.

6. Expanding the square of three terms

Write

$$(a+b+c{)}^2=(a+b+c)(a+b+c).$$

Multiplying each term gives

$$a^2+ab+ac+ba+b^2+bc+ca+cb+c^2.$$

Collect equal products:

$$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca.$$

7. Factorising by a common factor

First identify a factor shared by every term. For

$$ax+ay,$$

the common factor is $a$. Take it outside the brackets:

$$ax+ay=a(x+y).$$

This reverses the expansion of $a(x+y)$.

8. Factorising a difference of two squares

Recognise the pattern

$$a^2-b^2.$$

Use the identity

$$a^2-b^2=(a+b)(a-b).$$

Thus, a difference of two square terms factorises into the sum and difference of their square roots.

9. Factorising a perfect-square trinomial

Match the expression to one of the standard forms:

$$a^2+2ab+b^2=(a+b{)}^2$$

or

$$a^2-2ab+b^2=(a-b{)}^2.$$

The sign of the middle term determines whether the factor is a sum or a difference.

10. Factorising $x^2+(a+b)x+ab$

To factorise

$$x^2+(a+b)x+ab,$$

find two numbers whose sum is $a+b$ and whose product is $ab$. If those numbers are $a$ and $b$, then

$$x^2+(a+b)x+ab=(x+a)(x+b).$$

The result can be verified by expanding the factors.

11. Checking an identity

Substitute selected numerical values into both sides of the proposed identity. If both sides produce the same value, the example is consistent with the identity. However, this numerical check does not prove the identity for every permissible value; an algebraic derivation is required for a general proof.

Where It Goes Wrong

• Applying the distributive property to only one term in a bracket instead of multiplying every term.
• Losing or changing signs when expanding expressions containing subtraction or negative terms.
• Confusing $(a+b{)}^2$ with $a^2+b^2$ and omitting the required $2ab$ term.
• Failing to combine the two middle terms when expanding a square or two binomials.
• When factorising, not first arranging terms suitably or checking for a common factor.
• Using a numerical check as if it proved an identity for all permissible values; algebraic proof is required.

What Gets Asked

• Define an algebraic expression, term, coefficient, identity, expansion, factorisation, common factor, visual area model, or polynomial.
• Expand a single bracket using

$$a(b+c)=ab+ac$$ or $$a(b-c)=ab-ac.$$

• Expand products of binomials, including

$$(x+a)(x+b).$$

• Use the square-of-a-sum, square-of-a-difference, sum-and-difference, or three-term identities.
• Explain or use the area model for

$$(a+b{)}^2$$ with regions $a^2$, $ab$, $ab$, and $b^2$.

• Factorise an expression by taking out a common factor.
• Factorise a difference of two squares.
• Identify and factorise perfect-square trinomials.
• Factorise

$$x^2+(a+b)x+ab$$ by finding two numbers with the required sum and product.

• Prove an identity algebraically or distinguish an algebraic proof from a numerical check.
• Explain why expansion and factorisation are inverse processes and why equivalent expressions can have different appearances but the same value.