Download Test #1 with the Key

Mathematics 1065-300
College Algebra
Summer Session II, 2010
Test #1
Instructor: Alexandra Shlapentokh
(1) a 2 − b 2 =
(a) a − b
(b) (a − b)2
(c) (a − b)(a + b)
(d) (a + b)2
(e) None of the above
(2) a 2 + b 2 =
(a) a − b
(b) (a − b)2
(c) (a − b)(a + b)
(d) (a + b)2
(e) None of the above
(3) (a + b)2 =
(a) a 2 + b 2
(b) a 2 + 2ab + b 2
(c) a 2 − 2ab + b 2
(d) (a + b)(a − b)
(e) None of the above
(4) (a − b)2 =
(a) a 2 + b 2
(b) a 2 + 2ab + b 2
(c) a 2 − 2ab + b 2
(d) (a + b)(a − b)
(e) None of the above
(5) a 3 − b 3 =
(a) (a − b)3
(b) (a − b)(a 2 + ab + b 2 )
(c) (a + b)(a 2 − ab + b 2 )
(d) (a + b)(a 2 − b 2 )
(e) None of the above
(6) a 3 + b 3 =
(a) (a − b)3
(b) (a − b)(a 2 + ab + b 2 )
(c) (a + b)(a 2 − ab + b 2 )
(d) (a + b)(a 2 − b 2 )
(e) None of the above
(7) (a − b)3 =
(a) a 3 − b 3
(b) (a − b)(a 2 − ab + b 2 )
1
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(c) (a + b)(a 2 − ab + b 2 )
(d) (a + b)(a 2 − b 2 )
(e) None of the above
(a + b)3 =
(a) a 3 − b 3
(b) a 3 − 3a 2 b + 3ab 2 − b 3
(c) (a + b)(a 2 − ab + b 2 )
(d) (a + b)(a 2 − b 2 )
(e) None of the above
What is the degree of the polynomial x 4 + x + x 3 + 1
(a) 1
(b) 2
(c) 3
(d) 4
(e) None of the above
(5x 3 + 2x 2 − 3x + 2) + (−3x 3 − 2x 2 + 3x − 2) =
(a) 2x 3 + 2
(b) 2x 3 + 6x + 4
(c) 8x 3 + 4x 2 + 6x + 4
(d) 2x 3
(e) None of the above
x − 56 x =
(a) 46 x
(b) This expression cannot be simplified.
(c) 16 x
(d) − 61 x
(e) None of the above
(x + 1)(x 2 + x + 1) =
(a) x 3 + 2x 2 − 2x + 1
(b) x 3 + 2x 2 + 2x + 1
(c) x 3 + 2x 2 + x + 1
(d) x 3 + x 2 + x + 1
(e) None of the above
(x + 2)(x + 3) =
(a) x 2 + 5
(b) x 2 + 5x + 6
(c) x 2 − 5x + 5
(d) x 2 + 6
(e) None of the above
If a polynomial of degree 5 is divided by a polynomial of degree 3, then which of the
following cannot be the degree of the remainder:
(a) 0
(b) 1
(c) 2
(d) 3
2
(e) None of the above
(15) What is the remainder from the division of x 3 + x 2 + x + 6 by x + 1?
(a) 2x + 3
(b) 3
(c) 5
(d) 1
(e) None of the above
(16) Suppose b(x) is a non-zero polynomial of degree greater than 2 and
x 5 + x + 1 = b(x)(x 2 + x + 1) + (x 2 + 1).
(17)
(18)
(19)
(20)
(21)
What is the remainder from the division of x 5 + x + 1 by b(x)?
(a) Cannot be determined from the information given in the problem
(b) x 2 + 1
(c) x 2 + x + 1
(d) x 5 + x + 1
(e) None of the above
What is the quotient from the division of x 3 + 2x 2 + x + 5 by x + 2?
(a) x 2 + 1
(b) x + 1
(c) x 2 − 1
(d) 2
(e) None of the above
One of the factors in the complete factorization of x 2 − 4 over integers is
(a) x
(b) x − 2
(c) x + 3
(d) x 2 + 4
(e) None of the above
One of the factors in the complete factorization of x 3 − 8 over integers is
(a) x
(b) x − 2
(c) x + 3
(d) x 2 + 4
(e) None of the above
One of the factors in the complete factorization of x 2 + 5x + 6 over integers is
(a) x
(b) x − 2
(c) x + 3
(d) x 2 + 4
(e) None of the above
One of the factors in the complete factorization of 2x 2 + 3x + 1 over integers is
(a) 2x + 1
(b) x − 1
(c) x + 3
(d) x 2 + 4
(e) None of the above
3
(22) One of the factors in the complete factorization of x 3 − x 2 + x − 1 over integers is
(a) 2x + 1
(b) x − 1
(c) x + 3
(d) x 2 + 4
(e) None of the above
1
3
(23) Solve 3x−6
= x+2
(a) no solutions
(b) x = −2
(c) x = 11
(d) x = 26
(e) None of the above
(24) How many real solutions does the equation x 2 + x + 10 = 0 have?
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of the above
(25) How many real solutions does the equation x 2 + 2x + 1 = 0 have?
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of the above
(26) One of p
the solutions of x 2 + 5x − 1 = 0 is
5+ 29
(a)
2
(b) −5
2
(c)
p
−5+ 29
2p
−5+ 27
2
(d)
(e) None of the above
(27) Given a quadratic polynomial ax 2 +bx+c with a 6= 0, b, c being real numbers, completing
the square means rewriting the polynomial in the form
(a) ax 2 + bx + c = a(x + h) + k, for some real numbers h and k.
(b) ax 2 + bx + c = a(x + h)2 + k, for some real numbers h and k.
(c) ax 2 + bx + c = a(x + h)2 + (xk)2 , for some real numbers h and k.
(d) ax 2 + bx + c = a(x + h)2 − x 2 , for some real numbers h and k.
(e) None of the above
(28) Completing the square for the polynomial x 2 + 2x − 1 will result in an expression of the
form
(a) (x − 1)2 + k, for some real number k < 0.
(b) (x + 1)2 + k, for some real number k < 0.
(c) (x + 1)2 + k, for some real number k > 0.
(d) One cannot complete the square for this polynomial.
(e) None of the above
(29) Solve (x − 3)2 = 7,
4
(30)
(31)
(32)
(33)
(34)
(35)
(36)
p
(a) 3 + p7
(b) 3 ± p7
(c) 3p
− 7
(d) 3 7
(e) None of the above
Completing the square for the polynomial x 2 + 4x − 1 will result in an expression of the
form
(a) (x + h)2 + 5, for some real number h > 0.
(b) (x + h)2 − 5, for some real number h > 0.
(c) (x + h)2 − 5, for some real number h < 0.
(d) One cannot complete the square for this polynomial.
(e) None of the above
1252/3 43/2 =
(a) 10
(b) 20
(c) 30
(d) 40
(e) None of the above
a 2/3 b −1/2
Simplify −1/3 1/2
a
b
(a) a
(b) b
(c) ab
(d) ba
(e) None of the above
Principal square root of a non-negative real number
(a) is always a positive real number.
(b) is always a negative real number.
(c) is not always a real number.
(d) can be positive or negative.
(e) None of the above
Principal cube root of a real number
(a) is always a positive real number.
(b) is always a negative real number.
(c) is not always a real number.
(d) can be positive or negative.
(e) None of the above
p
For a positive
real number a we have that 12a 3 =
p
(a) 2a p3a
(b) 3a p2a
(c) 4a 2p 3a
(d) 2a 3a 2
(e) None of the above
How many real solutions does the equation x 2 + x − 10 = 0 have?
(a) 0
(b) 1
5
(c) 2
(d) 3
(e) None of the above
6
Key
1c, 2e, 3b, 4c, 5b, 6c, 7e, 8e, 9d, 10d, 11c, 12b, 13b, 14d, 15c, 16b, 17a, 18b, 19b, 20c, 21a, 22b,
23a, 24a, 25b, 26c, 27b, 28b, 29b, 30b, 31e, 32d, 33a, 34d, 35a, 36c.
7