ALGEBRA - solved problems
Algebraic expressions
Simplifying algebraic expressions
 1 Simplify algebraic expressions.
 Solutions: a)   - 4a3 + 3a2 + 5a3 - 7a2 = (- 4 + 5) · a3 + (3 - 7) · a2  = a3 - 4a2, b)   (x2  - x + 1) · (x + 1) = x3 - x2 + x + x2  - x + 1 = x3 + 1.
Evaluating algebraic expressions
 2 Evaluate the expression  x2  - 6xy + 9y2  for x = 2  and  y =  -1.
 Solution: x2  - 6xy + 9y2 = 22  - 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 = x4  - 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 + 6a3b3  = 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  + (-2)3 = x3  - 6x2 + 12x - 8, c)  (2x + y)3 = (2x)3 + 3 · (2x)2 · y + 3 · (2x) · y2  + y3 = 8x3 + 12x2y + 6xy2  + 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 - y2  = y · (x - y),     c)  a - a2  = a · (1 - a), d)  x3 -3x2  + x = x · (x2 - 3x +1),    e)  x(a + b) - (a + b) = (a + b) · (x - 1), f)   a(x - 3y) - x + 3y  = 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 = 12  - 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 [y2  - (2x)2] = 5y(y - 2x)(y + 2x), c)   9x2  - (x + 2)2 = [3x - (x + 2)] · [3x + (x + 2)] = (2x -2) · (4x + 2) = 4(x -1) · (2x +1).
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