Here are the core algebraic formulas you’ll use most:

1 Squares
– (a + b)² = a² + 2ab + b²
– (a+b)² = (a-b)² + 4ab
– (a – b)² = a² – 2ab + b²
– (a-b)²= (a+b)² – 4ab
– a² – b² = (a + b)(a – b)

2 Cubes
– (a + b)³ = a³ + 3a²b + 3ab² + b³
– (a + b)³ = a³ + b³ + 3ab(a + b)
– (a – b)³ = a³ – 3a²b + 3ab²- b³
– (a – b)³ = a³ – b³ – 3ab(a – b)
– a³ + b³ = (a + b)(a² – ab + b²)
– a³ – b³ = (a – b)(a² + ab + b²)

1. If f(x) = 4x³ – 2x² + 5x – 8 is divided by x – 2, then what will be the remainder?

• A) 26
• B) 42
• C) 50
• D) 34

Show Answer
Ans: A) 26
Solution:
This is a remainder theorem problem.
x-2 = 0 => x = 2
Put the value of x into 4x³ – 2x² + 5x – 8 OR

2. $\frac{a}{b}+\frac{b}{a}$=-1 then,What is the value of a³-b³ ?

• a) 1
• b) -1
• c) ½
• d) 0

Show Answer
Ans: D) 0
Solution : $\frac{a}{b}+\frac{b}{a}$=-1
$\frac{\mathrm{a²+\; b²}}{\mathrm{ab}}$= -1
=> a²+ab+b²=0
a³-b³= (a-b)(a²+ab+b²)
(a-b)× 0 = 0

3. If x + $\frac{1}{x}$= 7 then,What is the value of x³ + $\frac{1}{\mathrm{x³}}$ ?

• a) 343
• b) 322
• c) 364
• d) 336

Show Answer
Ans: b) 322
(a + b)³ = a³ + b³ + 3ab(a + b)
Solution : ( x + $\frac{1}{x}$)³ = x³ + $\frac{1}{\mathrm{x³}}$ +3(x + $\frac{1}{x}$)
=> 7³ = x³ + $\frac{1}{\mathrm{x³}}$+ 3 × 7
=> 343 = x³ +$\frac{1}{\mathrm{x³}}$ + 21

4. If $\frac{x}{y}$ = $\frac{4}{5}$ and 7x + 6y = 29, what is the value of y ?

• a) 2.4
• b) 2.5
• c) 2.3
• d) 1.3

Show Answer
Ans: b) 2.5
Solution :
Let the value of x be 4k.
Let the value of y be 5k.
7x + 6y = 29
28k + 30k = 29
58k = 29
.°. k = $\frac{1}{2}$