ALGEBRA - solved problems
Sum and difference of cubes
91.    Factorize given sum and difference of cubes.
Solutions:   a)  x3 + 8 = x3 + 23 = (x + 2) (x2 - 2x + 22),
                        since  (x + 2)(x2 - 2x + 4) = x3 - 2x2 + 4x + 2x2 - 4x + 8 = x3 + 8,
b) 8a3 -125 = (2a)3 - 53 = (2a - 5) [(2a)2 + (2a)5 + 52] = (2a - 5)(4a2 + 10a + 25),
  since  (2a - 5)(4a2 + 10a + 25) = 8a3 + 20a2 + 50a - 20a2 - 50a -125 = 8a3 -125.
Using a variety of methods including combinations of the above to factorize expressions
92.    Factorize given expressions.
Solutions:   a)  x2 - 2xy + y2 + 2y - 2x = (x - y)2 - 2(x - y) = (x - y)(x - y - 2),
b)  x2 - y2 + xz - yz = (x - y)(x + y) + z(x - y) = (x - y)(x + y + z),
c)  4x- 4xy  + y2  - z2 = (2x - y)2   - z2 = (2x - y - z)(2x - y + z),
d)  a- 7a + 6 = a- a - 6a + 6 = a(a2 -1) - 6(a -1) = (a -1)[a(a + 1) - 6] = (a -1)(a2 + a - 6)  
                             = (a -1)(a2 + 3a - 2a - 6) = (a -1)[a(a + 3) - 2(a + 3)] = (a -1)(a + 3)(a - 2).
  Factoring and expanding algebraic expressions, rules for transforming algebraic expressions
Expanding algebraic expressions
The square of a binomial, a perfect square trinomial
  (a + b)2 = a2 + 2ab + b2,
  (a - b)2a2 - 2ab + b2,
The square of a trinomial
  (a - b + c)2 = a2 + b2 + c2 - 2ab + 2ac - 2bc,
The cube of a binomial
  (a + b)3 = a3 + 3a2b + 3ab2 + b3,
  (a - b)3 = a3 - 3a2b + 3ab2 - b3,
Factoring algebraic expressions
Difference of two squares
  x2 - y2 = (x - y) (x + y),
Sum and difference of cubes
  x3 - y3 = (x - y) (x2 + xy + y2),
  x3 + y3 = (x + y) (x2 - xy + y2).
The sum and/or difference of any two numbers raised to the same (positive integer) power
x4 - y4 = (x - y) (x3 + x2y + xy2 + y3) = (x2 - y2) (x2 + y2
x2n - y2n = (x - y) (x2n-1 + x2n-2y + ¼  + xy2n-2 + y2n-1) = (xn - yn) (xn + yn
The binomial expansion algorithm - the binomial theorem
The binomial expansion of any positive integral power of a binomial, which represents a polynomial with n + 1 terms, 
or written in the form of the sum formula
 
 
is called the binomial theorem.
The binomial coefficients can also be obtained by using Pascal's triangle.
The triangular array of integers, with 1 at the apex, in which each number is the sum of the two numbers above it in the preceding row, as is shown in the initial segment in the diagram, is called Pascal's triangle.
So, for example the last row of the triangle contains the sequence of the coefficients of a binomial of the 5th power.
n 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
- 1 - - - - - 1
93.    Expand the binomial using the binomial theorem.
Solution:
 
 
 
 
 
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