Exploring Algebraic Identities · Lesson 8 of 8

Chapter Summary and Practice

“Expand, factor and prove that both directions eventually lead home.”

Learning Objectives

• Recall the major identities of the chapter. • Choose suitable identities for expansion, factorisation and calculation. • Apply splitting-the-middle-term methods confidently. • Use cubic identities and simplify rational expressions. • Solve mixed problems that combine several chapter ideas.

This chapter developed identities as algebraic statements that remain true for all allowed values. We visualised them through areas and volumes, used them for quick calculation, reversed them for factorisation, and extended the same reasoning to quadratics, cubics and rational expressions.

Key Square Identities

Square of a sumLaTeX

(x + y)^{2} = x^{2} + 2 x y + y^{2}

Square of a differenceLaTeX

(x - y)^{2} = x^{2} - 2 x y + y^{2}

Square of three termsLaTeX

(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 z x

Difference of squaresLaTeX

x^{2} - y^{2} = (x + y) (x - y)

Product Identities

Monic linear factorsLaTeX

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

General linear factorsLaTeX

(a x + b) (c x + d) = a c x^{2} + (a d + b c) x + b d

Cubic Identities

Cube of a sumLaTeX

(x + y)^{3} = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}

Cube of a differenceLaTeX

(x - y)^{3} = x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3}

Difference of cubesLaTeX

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

Sum of cubesLaTeX

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

Three-variable cubic identityLaTeX

x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - x z - y z)

How to Choose the Right Method

Pattern	First idea
Two terms squared	Use (a±b)²
Three terms squared	Use (a+b+c)²
Difference of perfect squares	Use a²−b²
x²+Bx+C	Split middle term using sum B and product C
Ax²+Bx+C	Use A×C when splitting
Perfect cube pattern	Use (a±b)³
Sum or difference of cubes	Use cube factorisation identities
Rational algebraic expression	Factor numerator and denominator first

Common Mistakes

• Writing (a+b)²=a²+b², • Losing the negative middle term in (a−b)², • Choosing numbers that satisfy the product but not the required sum, • Cancelling terms instead of factors, • Forgetting denominator restrictions, • Confusing x³+y³ with (x+y)³, • Missing a common factor before applying an identity,

Guided Practice

Guided Example 1: Three-Term Square

Problem
Expand (3x−2y+4z)².

1.Square each term: 9x²+4y²+16z².
2.Double pairwise products: −12xy, −16yz and +24xz.
3.Result: 9x²+4y²+16z²−12xy−16yz+24xz.

Guided Example 2: Perfect Square

Problem
Factor 49g²+14gh+h².

1.49g²=(7g)².
2.h² is the second square.
3.14gh=2(7g)(h).
4.Therefore the factorisation is (7g+h)².

Guided Example 3: Split the Middle Term

Problem
Factor 6x²+7x+2.

1.A×C=12.
2.Choose 3 and 4.
3.6x²+3x+4x+2.
4.=3x(2x+1)+2(2x+1).
5.=(3x+2)(2x+1).

Guided Example 4: Difference of Cubes

Problem
Factor 27u³−1/125.

1.27u³=(3u)³ and 1/125=(1/5)³.
2.Use a³−b³=(a−b)(a²+ab+b²).
3.Result: (3u−1/5)(9u²+3u/5+1/25).

Guided Example 5: Rational Expression

Problem
Simplify (4x²+4x+1)/(4x²−1), assuming the denominator is non-zero.

1.Numerator=(2x+1)².
2.Denominator=(2x+1)(2x−1).
3.Cancel the common factor 2x+1 when it is non-zero.
4.Result: (2x+1)/(2x−1).

Practice Questions

Expand (−3x+4)².
Evaluate (2s+7)(2s−7) using an identity.
Expand (−3m+4k−l)².
Evaluate 17×21 using a suitable identity.
Evaluate 104×96 using a suitable identity.
Evaluate 199³ using a suitable identity.
Factor 9m²−1/(25n²).
Factor 27b³−1/(64b³).
Factor 64y³+z³/125.
Factor 9m²−12m+4.
Factor 4x²+9y²+36z²+12xz+36yz+24xy.
Simplify (4x²+4x+1)/(4x²−1), assuming the denominator is non-zero.
Find possible dimensions of a rectangle with area 25a²−30ab+9b².
A square playground has side 40 m and a path of width s m around it. Find an expression for the path area.
A rectangular pool has area 2x²+7x+3 and width 2x+1. Find its length.
If a+b+c=5 and ab+bc+ca=10, prove a³+b³+c³−3abc=−25.
Factor n³−n and explain why it is divisible by 6 for every natural n.

Quick check

Which expression equals (a+b)²?

Quick check

Which pair factors x²+11x+30?

Quick check

What is x³−y³?

Quick check

What may be cancelled in a rational algebraic expression?

Quick check

What makes an equation an identity?

Before You Finish the Chapter

Check that you can derive the main square identities, recognise perfect-square forms, factor quadratics by splitting the middle term, understand algebra tiles, use cubic identities, and simplify rational expressions by factorisation.

Key Takeaways

• Identities are true for all allowed values. • Geometric models explain identities through area and volume. • Identities support expansion, factorisation and fast calculation. • Quadratic factorisation depends on sum-product structure. • Cubic identities extend the same reasoning to higher powers. • Rational expressions simplify through factorisation and cancellation of common non-zero factors.

Coming Next

This completes Exploring Algebraic Identities. Continue by practising how to recognise structure before choosing the correct identity.

Previous · Lesson 7

Finding New Identities

End of chapter

Key Square Identities

Product Identities

Cubic Identities

How to Choose the Right Method

Guided Practice

Practice Problems

Quiz

Key Takeaways