Polynomials and fractions

2010000903

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

2010000904

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

2010001306

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

9000087502

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

9000087503

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{3} {2}\right \}\), find the quotient of the polynomials: \[ (x^{2} + x + 1) : (2x + 3) \]
\(\frac{1} {2}x -\frac{1} {4} + \frac{\frac{7} {4} } {2x+3}\)
\(\frac{1} {2}x -\frac{1} {2} + \frac{\frac{7} {4} } {2x+3}\)
\(x + 2 + \frac{7} {2x+3}\)
\(x - 2 + \frac{7} {2x+3}\)

9000087504

Level: 
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{3} {5}\right \}\), find the quotient of the polynomials. \[ (5x^{3} - 2x^{2} + x + 1) : (5x + 3) \]
\(x^{2} - x + \frac{4} {5} - \frac{\frac{7} {5} } {5x+3}\)
\(x^{2} - x + \frac{4} {5} + \frac{\frac{7} {5} } {5x+3}\)
\(x^{2} - x + \frac{4} {5} - \frac{\frac{9} {5} } {5x+3}\)
\(x^{2} - x + \frac{4} {5} + \frac{\frac{9} {5} } {5x+3}\)