ALGEBRA - solved problems
  Algebraic expressions
Simplifying algebraic expressions
1.    Simplify algebraic expressions.
Solutions:   a)   - 4a3 + 3a2 + 5a3 - 7a2 = (- 4 + 5) a3 + (3 - 7) a = a3 - 4a2,
b)   (x- x + 1) (x + 1) = x3 - x2 + x + x2  - x + 1 = x3 + 1.
Evaluating algebraic expressions
2.    Evaluate the expression  x- 6xy + 9y2  for x = 2  and  y -1.
Solution: x- 6xy + 9y2 = 2- 6 2 (- 1)  + 9 (-1)2 = 4 + 12 + 9 = 25.
Expanding algebraic expression by removing parentheses ( brackets)
3.    Expand given expressions.
Solutions:   a)   (a - b)2 = (a - b) (a - b) = a2 - ab - ab + b2 = a2 - 2ab + b2,
b)   (a - b) (a + b) = a2 - ab + ab - b2 = a2 - b2,
c)   (x + y) (x2 - xy + y2) = x3 - x2y + xy2 + x2y  - xy2 + y3 = x3 + y3.
The square of a binomial (or binomial square)
4.    Square given binomials.
Solutions:   a)   (a + b)2 = (a + b) (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2,
b)   (2x + 3)2 = (2x)2 + 2 (2x) 3 + 32 = 4x2 + 12x + 9,
c)   (x - 2y)2 = x2  + 2 x (-2y) + (-2y)2 = x2 - 4xy + 4y2.
Squaring trinomial (or trinomial square)
5.    Square given trinomials.
Solutions:   a)  (x2 - 2x + 5)2 = (x2)2 + (2x)2 + 52 + 2 x2 (-2x) + 2 x2 5 + 2 (-2x) 5 =
                           = x4 + 4x2 + 25 - 4x3 + 10x2 - 20x = x - 4x3 + 14x2 - 20x + 25,
b)  (a3 - a2b - 3ab2)2 = (a3)2 + (a2b)2 + (3ab2)2 + 2a3 (-a2b) + 2a3 (-3ab2) + 2(-a2b) (-3ab2) =
    = a6 + a4b2 + 9a2b4 - 2a5b - 6a4b2 + 6a3b= a6 - 5a4b2 + 9a2b4 - 2a5b + 6a3b3.
Cube of a binomial
6.    Cube (rise to third power) given binomials.
Solutions:   a)  (a - b)3 = (a - b)2 (a - b) = (a2 - 2ab + b2) (a - b)
                  = a3 - 2a2b + ab2 - a2b + 2ab2 - b3 = a3 - 3a2b + 3ab2 - b3,
b)  (x - 2)3 = x3 + 3 x2 (-2) + 3 x (-2)+ (-2)3 = x- 6x2 + 12x - 8,
c)  (2x + y)3 = (2x)3 + 3 (2x)2 y + 3 (2x) y+ y3 = 8x3 + 12x2y + 6xy+ y3.
  Factoring algebraic expressions
Factoring algebraic expression by finding (determining) a common factor
7.    Factorize given expressions.
Solutions:   a)  3x - 6y = 3 (x - 2y),     b)  xy - y = y (x - y),     c)  a - a = a (1 - a),
d)  x3 -3x+ x = x (x2 - 3x +1),    e)  x(a + b) - (a + b) = (a + b) (x - 1),
f)   a(x - 3y) - x + 3 = a(x - 3y) - (x - 3y) = (x - 3y) (a - 1).
Grouping like terms, grouping and factorizing four terms
8.    Factorize given expressions.
Solutions:   a)  ax - bx - a + b = x(a - b) - (a - b) = (a - b) (x - 1),
b)  a - 1 - ab + b = (a - 1) - b (a - 1) = (a - 1) (1 - b),
c)  x2 + ax - bx - ab = x(x + a) - b (x + a) = (x + a) (x - b),
d)  5ab2 - 3a3 - 10b3 + 6a2b = 5b2(a - 2b) -3a2(a - 2b) = (a - 2b)(5b2 - 3a2).
The square of a binomial - perfect squares trinomials
9.    Factorize given expressions.
Solutions:   a)  1 - 4x + 4x2 = 1- 2 2x + (2x)2 = (1 - 2x)2  = (1 - 2x) (1 - 2x),
b)  a5 + 6a4b + 9a3b2 = a3 (a2  + 6ab  + 9b2 ) = a3(a + 3b)2 = a3(a + 3b)(a + 3b).
Difference of two squares
10.    Factorize given expressions.
Solutions:   a)  16x2 - 1 = (4x)2 - 12 = (4x -1) (4x +1),
b)  5y3 - 20x2y = 5y (y2 - 4x2) = 5y [y - (2x)2] = 5y(y - 2x)(y + 2x),
          c)   9x- (x + 2)2 = [3x - (x + 2)] [3x + (x + 2)] = (2x -2) (4x + 2) = 4(x -1) (2x +1).
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