Expansions
ICSE Class 9 · Mathematics · Chapter 4
What this chapter covers
- Products, expansions and identities
- The square of a binomial: (a ± b)2
- The cube of a binomial: (a ± b)3
- The product (x ± a)(x ± b)
- The square of a trinomial: (a ± b ± c)2
- Finding values using the identities
- Using expansions for quick calculation
- Common mistakes and how to avoid them
Key points
- An expansion writes a product of expressions as a sum of terms with like terms collected; an identity is true for all values of its variables.
- (a ± b)2 = a2 ± 2ab + b2; the middle term 2ab is never zero unless a or b is zero.
- (a + b)2 + (a - b)2 = 2(a2 + b2) and (a + b)2 - (a - b)2 = 4ab.
- (a ± b)3 = a3 ± 3a2b + 3ab2 ± b3, or in compact form a3 ± b3 ± 3ab(a ± b).
- (x + a)(x + b) = x2 + (a + b)x + ab; treat a and b as signed numbers to cover every sign case.
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca; a cross product is negative only when exactly one of its two terms is negative.
- For x and (1)/(x) the product is 1: (x ± (1)/(x))2 = x2 ± 2 + (1)/(x2) and (x ± (1)/(x))3 = x3 ± (1)/(x3) ± 3(x ± (1)/(x)).
- To find a value from given conditions, pick the identity that links the known expressions to the required one; do not try to find each variable.
- A value obtained from a square root carries ± unless the question fixes the sign.
- Writing a number as a round number plus or minus a small part lets the identities do quick arithmetic, such as 982 = 9604.
Questions answered
Define an algebraic identity. How does it differ from an equation?
An identity states that two expressions are equal for every value of the variables, such as (a + b)2 = a2 + 2ab + b2. An equation, such as 2x + 1 = 7, holds only for particular values of the variable (here x = 3).
Expand (3a - 4b)2.
(3a)2 - 2(3a)(4b) + (4b)2 = 9a2 - 24ab + 16b2.
Expand ((x)/(2) + (2)/(x))2.
The product of the two terms is 1, so the result is (x2)/(4) + 2 + (4)/(x2).
Expand (x - 2y)3.
x3 - 3x2(2y) + 3x(2y)2 - (2y)3 = x3 - 6x2y + 12xy2 - 8y3.
Expand (y + 8)(y - 5) and (3x - 2)(3x - 7).
Sum of 8 and -5 is 3, product -40: y2 + 3y - 40. For the second, the sum of -2 and -7 is -9 and the product is 14: 9x2 - 27x + 14.
Expand (a - 2b - 3c)2.
a2 + 4b2 + 9c2 - 4ab + 12bc - 6ca. The term 12bc carries a plus sign because (-2b)(-3c) is positive.
Simplify (2x + 5)2 - (2x - 5)2.
Using (a + b)2 - (a - b)2 = 4ab, the result is 4(2x)(5) = 40x.
Why is (a + b)2 not equal to a2 + b2? Give a numerical example.
Multiplying (a + b)(a + b) produces two cross terms, ab and ba, which add to 2ab. With a = 2 and b = 3, (a + b)2 = 25 but a2 + b2 = 13; the difference is 2ab = 12.
Past board exam questions
ICSE 2019 sample paper · 3 marks
If x - (1)/(x) = 3, Evaluate x3 - (1)/(x3)
Show answer
x3 - (1)/(x3) = (x - (1)/(x))3 + 3(x - (1)/(x))
= 33 + 3 × 3 = 27 + 9 = 36