Polynomials and fractions

2010001306

Level: 
C
Factor the following polynomial. \[ x^{8} - 1 \]
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + 1\right )\left (x^{4} + 1\right )\)
\(\left (x - 1\right )^2\left (x + 1\right )^2\left (x^{2} + 1\right )^2\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{3} + 1\right )^2\)
\(\left (x - 1\right )^2\left (x + 1\right )^2\left (x^{4} + 1\right )\)

2010001305

Level: 
B
Factor the following polynomial expression. \[ 36b^{2}c^{2} - 9a^{2}b^{2} - 36c^{2}d^{2} + 9a^{2}d^{2} \]
\(9\left (b - d\right )\left (b + d\right )\left (2c + a\right )\left (2c - a\right )\)
\(\left (b^2 + d^2\right )\left (36c^2 + 9a^2\right )\)
\(9\left (a - d\right )\left (a + d\right )\left (2b + c\right )\left (2b - c\right )\)
\(\left (a^2 + d^2\right )\left (36b^2 + 9c^2\right )\)

2010000905

Level: 
B
Suppose we are given the following equality of two fractions with nonzero denominators. From the given expressions, choose the one that by substituting to the starred position makes the equality true. \[ \frac{2- 3x} {x +2} = \frac{2(9x^{2} - 12x + 4)} {*} \]
\((2x +4)(2 - 3x)\)
\((x +2)(2 - 3x)\)
\((x +2)(4 - 9x)\)
\((2x +4)(3x - 2)\)

2010000904

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{2} {3}\right \}\), find the quotient of the polynomials: \[ (x^{2} - x - 1) : (3x + 2) \]
\(\frac{1} {3}x -\frac{5} {9} + \frac{\frac{1} {9} } {3x+2}\)
\(\frac{1} {3}x -\frac{5} {9} - \frac{\frac{19} {9} } {3x+2}\)
\(\frac{1} {3}x -\frac{1} {9} + \frac{\frac{7} {9} } {3x+2}\)
\(\frac{1} {3}x -\frac{1} {9} - \frac{\frac{11} {9} } {3x+2}\)

2010000903

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{\pm 1\right \}\), find the quotient of the polynomials: \[ (-3x^{4} + 2x^{2} -4) : (x^{2} + 1) \]
\(- 3x^{2} + 5 - \frac{9} {x^{2}+1}\)
\(- 3x^{2} - 5 - \frac{9} {x^{2}+1}\)
\(- 3x^{2} + 5 +\frac{1} {x^{2}+1}\)
\(- 3x^{2} - 5 +\frac{1} {x^{2}+1}\)