# Rational Expressions Problems and Formulas for Short Multiplication

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 monomial?
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
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

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) –xp.xq
C) 3.(xp).y.x.(yp)
D)(x3m/xm)/xm
E) xpyq/(x4-p.y-4+q)
Solution:
A) –a2.a6 = -a2 + 6 = -a8
B) -xp.xq = -xp + q
C) 3xp.y.x.yp = 3.xp + 1.y1 + p
D) [x3m/xm] / xm = [x(3m - m)] / xm = x2m / xm = x(2m - m) = xm
E) xp.yq/[(x(4 - p)).y(-4 + q)] = xp - (4 - p).yq - (-4 + q) = xp - 4 + p.yq + 4- q = x2p - 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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