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Rational Expressions Problems and Formulas for Short Multiplication

Additonal problems

Problem 1 What about the rational expression is it a whole number according to x.
A) x² - 9x + 2
B) (3x + 2 )/(a - 1)
C) 2/3x² - 1/7x + 1/5a
D) (7 - x)/(x + 1)
E) x²-1/x
F) (1/2)x + 2/x
Solution:
The definition says that a rational expression is a whole one as to x only if x does not participate as a divisor. Then the expressions from A, B, C are whole ones, and D, E, F are fractional as to x.

Problem 2 Polynomial or term?
A) ab²x
B) ab(-3)c
C) x³ + x
D) –a + b
E) 2(a + b)³
F) 0.67ab
Direction: See the definition of term and polynomial.

Problem 3 Is the following monomial in normal form?
A) -3abc
B) x²y
C) (0.4)a.1/3b
D) x²yx
Answers: A and B - yes C and D - no

Problem 4 Represent the following monomial in normal form:
A)12xy²x²
B) 3x5ayx
C) 2.3/4ax²(-2/11)a²x
Solution: You must perform all marked operations and then you obtain: A) 12x³y²
B) 15ax²y
C) (-1/2)a³x³

Problem 5 What is the value of the term?:
A) -7x²y² when x = -2, y = 1
B) 7a²bc when a = 3, b = 2, c = 5/7
Solution:
A) -7.(-2)².(1)²= -7.4.1 = -28
B) 7.3².2.5/7 = 7.9.2.5/7 = 90

Problem 6 Represent each one of the expressions in normal form and determine its exponent as to the variables x and y
A) ax10xaax
B) ay.3/4.bxx.(-1/2)
C) 5cxy(-4)cxy
Answers: A) 10a³x³
B) -3/8.abx².y
C) -20c².x².y²
Exponents are: A) 3; B) 3; C) 4

Problem 7 Reduce the similar monomials:
A) 3x² - 5x² + 7(1/2)x²
B) (-5/7)a³b² - 2a³b² + 8a³b²
C) 2x - 5xy - 8xy - 3,1xy - 0,2xy
D) a³x² + 3a³x² + a²x³
Solution:
A) 9x - [-(5x - 1) - 8x] = 9x - (-5x + 1 - 8x) = 9x - (-13x + 1) = 9x + 13x - 1 = 22x - 1
B) 2a - {b - [a + (a - 2b)] + 3a} = 2a - {b - (a + a - 2b) + 3a} = 2a - (b - 2a + 2b + 3a) = 2a - (3b + a) = 2a - 3b - a = a - 3b
C)(x - y)(x² + xy + y²) + (x+y)(x² - xy + y²) = x³ - y³ + x³ + y³ = 2x³

Problem 8 What is the value of the polynomials:
A) 100x + 10y + z when x = 2, y = 3, z = 4
B) -0.08b + 73a²b + 27a²b when a = 0.2; b = 4
Answers: A) 234 B) 15.68

Problem 9 Write the following number as polynomial:
A) ab
B) abc
C) abba
D) ba
Instructions: Write the number through abc, in which c is the figure of the units, b is the figure of the e tens, and a is the figure of the hundreds. In this way we obtain:
A) ab = 10a + b
B) abc = 100a+10b+c
C) abba = 1000a + 100b + 10b + a = 1001a+10b
D) ba = 10b+a

Problem 10 Write the following expression in a normal polynomial:
A) (a²/4)² + (1/32)a(16a - 2a³)
B) 12x[y - (1/6)x] - 2y[(1/2)y + 3x]
Solution:
A) (a²/4)² + (1/32)a(16a - 2a³) = a⁴/16 + 16aa/32 - 2.a.a³/32 = a⁴/16 + a²/2 - a⁴/16 = a²/2
B) 12x[y - (1/6)x] - 2y[(1/2)y + 3x] = 12xy - 12x.(1/6)x - 2y.(1/2)y - 2y.3x = 12xy - 2x² - y² - 6xy = 6xy - 2x² - y²

Problem 11 Expand the following expression to multipliers:
A) a(c - b) + d(b - c)
B) a - 3b - 2x(3b - a) + 4y(3b - a)
C) 6(3 - b) + 9(b - 3)²
D) 5a(2x - 7) + 3b(7 - 2x)
Solution:
A) a(c - b) + d(b - c) = a(c - b) - d(c - b) = (c - b)(a - d)
B) a - 3b - 2x(3b - a) + 4y(3b - a) = a - 3b + 2x(a - 3b) - 4y(a - 3b) = (a - 3b)(1 + 2x - 4y)
C) 6(3 - b) + 9(b - 3)² = 6(3 - b) + 9(3 - b)² = (3 - b)[6 + 9(3 - b)] = (3 - b)(6 + 27 - 9b) = (3 - b)(33 - 9b) = 3(3 - b)(11 - 3b)
D) 5a(2x - 7) + 3b(7 - 2x) = 5a(2x - 7) - 3b(2x - 7) = (2x - 7)(5a - 3b)

Problem 12 Multiply the polynomials:
A) (x - y)(x + y)
B) (x + 5)(y - 7)
C) (2y - 7)(x + 1/y)
D) (-2a - 3b)(a + b³)
E) (a + b - c)(x - y)
Solution:
B) (x + 5)(y - 7) = xy + 5y + x(-7) + 5(-7) = xy + 5y - 7x - 35
D) (-2a - 3b)(a + b³) = (-2a).a - 3ba - 2a.b³ - 3bb³ = -2a² - 3ab - 2ab³ - 3b⁴
Answers: A) x² - y² C) ax + bx - cx - ay - by + cy

Problem 13 Perform the necessary operations and represent the polynomial to normal form:
A) (a + 1)(a² - a + 1)
B) (a - 7)(a + 2) - (2a - 1)(a - 14)
C) (2 - b)(1 + 2b) + (1 + b)(b³ + 3b)
D) (x - y)(x + y) + x² + 2xy + y²
D) –(x - y)² + x² - 2yx + y²
Solution:
From the shortened multiplication formulas given we have:
A) (a + 1)(a² - a + 1) = a³ + 1
B) (a - 7)(a + 2)-(2a - 1)(a - 14) = a.a - 7.a + 2.a - 7.2 - (2aa - 1a - 2a.14 + 1.14) = a²- 7a + 2a - 14 - (2a² - a - 28a + 14) = a² - 5a - 14 - (2a² - 29a + 14) = a² - 5a - 14 - 2a² + 29a - 14 = -a² + 24a - 28
C) (2 - b)(1 + 2b) + (1 + b)(b³ + 3b) = 2 + 4b - b - 2b² + b³ + 3b + b⁴ + 3b² = b⁴ + b³ + b² + 6b + 2
D) (x - y)(x + y) + x² + 2xy + y² = x² - y² + x²+ 2xy + y² = 2x² + 2xy
E) -(x - y)² + x² - 2xy + y² = -(x² - 2xy + y²)+x² - 2xy + y² = -x² + 2xy - y² + x² - 2xy + y² = 0

Problem 14 Perform the following operations:
A) [2x(-5xy)]/2xy
B) 21x³y²/[(-7x²y)/x⁵]
C) a⁸/[(a²/a) / (a²/a)]
Solution:
A) [2x(-5xy)]/2xy = (-10x²y)/2xy = -5x
B) 21x³y²/[(-7x²y)/x⁵] = 21x³y².x⁵/(-7x²y) = 21x⁸y²/(-7x²y) = -3x⁶y
C) a⁸/[(a²/a)/(a²/a)] = a⁸/[(a².a)/a.a²] = a⁸/(a³/a³) = a⁸/1 = a⁸

Problem 15 Perform the following multiplication:
A) –a².a⁶
B) –x^p.x^q
C) 3.(x^p).y.x.(y^p)
D)(x³m/x^m)/x^m
E) x^py^q/(x^4-p.y^-4+q)
Solution:
A) –a².a⁶ = -a^{2 + 6} = -a⁸
B) -x^p.x^q = -x^{p + q}
C) 3x^p.y.x.y^p = 3.x^{p + 1}.y^{1 + p}
D) [x^3m/x^m] / x^m = [x^{(3m - m)}] / x^m = x^2m / x^m = x^{(2m - m)} = x^m
E) x^p.y^q/[(x^{(4 - p)}).y^{(-4 + q)}] = x^{p - (4 - p)}.y^{q - (-4 + q)} = x^{p - 4 + p}.y^{q + 4- q} = x^{2p - 4}.y⁴

Problem 16 Perform the following division:
A) (x³ – x² + y.x²)/(x²)
B) (m.n.a² + 3.a.n.m² – 2.n.a²)/3.a.n
C) (-6x².y³ + 8.y.x³ – x.y)/(-2x)
Solution:
A) (x³ – x² + y.x²)/(x²) = x³/x² – x²/x² + y.x²/x² = x^(3-2) – x^(2-2) + y.x^(2-2) = x – x⁰ + y.x⁰ = x - 1 + y
B) ( m.n.a² + 3.a.n.m² – 2.n.a²)/3.a.n = m.n.a²/3.a.n + 3.a.n.m²/3.a.n – 2.n.a²/3.a.n = 1/3.m.n^(1-1).a^(2-1) + a^(1-1).m².n^(1-1) – 2/3.n^(1-1).a^(2-1) = 1/3.m.a + m² – 2/3a
C) (-6x².y³ + 8.y.x³ – x.y)/(-2x) = -6.x².y³/(-2x) – (8yx³)/(2x) + xy / 2x = 3x^(2-1).y³ – 4y.x^(3-1) + 1/2y.x^(1-1) = 3xy³ – 4yx² + 1/2y

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