Algebraic Identities Practice Questions with Detailed Solutions – Level 3

Algebraic Identities Level 3 Practice Set: Sums of Cubes and Conditional Results

Aptitude360's Algebraic Identities Level 3 set is a Quant Algebra practice quiz on multi-step identity problems. It covers higher powers found from a given sum, symmetric expressions in a, b and c, and large numerical expressions that collapse through the cube identities. Conditional results, surd substitutions and telescoping products are also part of the set.

Topics tested in this Level 3 algebraic identities set

• Higher powers of x plus its reciprocal, built from a given sum or difference
• The sum of cubes of a, b and c minus three times their product, including consecutive numbers
• Conditional results when a + b + c equals zero
• Sums of squares, sums of cubes and squared products from given symmetric sums
• Telescoping products and alternating differences of squares
• Surd values substituted into polynomial expressions
• Decimal expressions that simplify through the sum and difference of cubes

Key formulas for advanced algebraic identities

• Sum of three cubes: \( a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca) \)
• If \( a+b+c=0 \): \( a^{3}+b^{3}+c^{3}=3abc \)
• Square of a trinomial: \( (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(ab+bc+ca) \)
• Difference of cubes: \( a^{3}-b^{3}=(a-b)^{3}+3ab(a-b) \)
• Difference of squares: \( a^{2}-b^{2}=(a+b)(a-b) \)

Revision tips for multi-step identity problems

• Before expanding anything, check whether a + b + c is zero, since many expressions then reduce at once
• Write each factor of a telescoping product as a difference of squares and cancel terms in pairs
• When a variable equals a surd, rationalise to find its reciprocal first
• Rearrange a sum of squares equal to zero into perfect squares to fix each variable

Who should take this algebra set

Learners who already handle basic and intermediate identities comfortably and want tougher algebra practice for CAT, SSC, banking and other aptitude exams.