9000083606
Level:
C
Assuming \(x\neq 2\),
simplify the expression
\(
\frac{x^{2} + x - 6}
{x^{3} - 8}.
\)
\(\frac{x+3}
{x^{2}+2x+4}\)
\(\frac{x+3}
{x^{2}-2x+4}\)
\(\frac{x+3}
{x^{2}+4x+4}\)
\(\frac{x+3}
{x^{2}-4}\)
Polynomials and Rational Expressions
9000083607
Level:
B
Assuming \(x\neq 0\),
\(x\neq \pm 1\),
\(y\neq 0\),
simplify the expression.
\[
\left [\left ( \frac{x}
{x + 1}\right )^{2} : \left (\frac{x - 1}
{y} \right )^{2}\right ] : \frac{2xy}
{x^{2} - 1}
\]
\(\frac{xy}
{2\left (x^{2}-1\right )}\)
\(4\)
\(\frac{x^{2}-1}
{4} \)
\(\frac{x-1}
{4} \)
9000083608
Level:
B
Assuming \(xy\neq - 1\),
simplify the expression:
\[
\frac{ \frac{x-y}
{1+xy} + y}
{1 -\frac{y(x-y)}
{1+xy} }
\]
\(x\)
\(\frac{x(1+y^{2})}
{1-y^{2}} \)
\(x - 1\)
\(x(1 + y^{2})\)
9000083610
Level:
B
Assuming \(x\neq \pm y\)
and \(y\neq 2x\),
simplify the expression:
\[
\left ( \frac{2x}
{x + y} + \frac{y}
{x - y} - \frac{y^{2}}
{x^{2} - y^{2}}\right ) : \left ( \frac{1}
{x + y} + \frac{x}
{x^{2} - y^{2}}\right )
\]
\(x\)
\(2x - y\)
\(\frac{x}
{2x-y}\)
\(1\)
9000079204
Level:
B
Find the domain of the following expression.
\[
\frac{x^{2} - x}
{x + 1} : \frac{x^{2} - 1}
{x^{2} + 2x + 1}
\]
\(\mathbb{R}\setminus \{ - 1;1\}\)
\(\mathbb{R}\setminus \{ - 1;0;1\}\)
\(\mathbb{R}\setminus \{ - 1\}\)
\(\mathbb{R}\setminus \{ - 1;0\}\)
9000079201
Level:
A
Evaluate
\[
\frac{-x^{2}}
{x - y} -\frac{y - x}
{x + y}
\]
at \(x = -1\),
\(y = 2\).
\(-\frac{8}
{3}\)
\(-\frac{10}
{3} \)
\(-\frac{2}
{3}\)
\(-\frac{4}
{3}\)
9000079202
Level:
B
Find the set \(M\)
of all the real \(x\)
for which the following expression is not a well defined number.
\[
\frac{x - 4}
{x^{3} - 16x}
\]
\(M = \{ - 4;0;4\}\)
\(M = \{ - 4;4\}\)
\(M = \{0;4\}\)
\(M = \{0\}\)
9000079210
Level:
A
Consider the expression
\[
V (x) = \frac{x}
{x - 1} - \frac{1}
{1 - x}.
\]
Find the correct ordering of the values
\(V (-2)\),
\(V (0)\) and
\(V (2)\).
\(V (0) < V (-2) < V (2)\)
\(V (-2) < V (0) < V (2)\)
\(V (0) < V (2) < V (-2)\)
\(V (2) < V (0) < V (-2)\)
9000079206
Level:
A
Assuming \(x\neq 0\),
\(y\neq 0\),
\(x\neq y\),
simplify the following expression.
\[ {
\frac{1}
{x^{2}} - \frac{1}
{y^{2}}
\over -\frac{1}
{y} + \frac{1}
{x}}
\]
\(\frac{x+y}
{xy} \)
\(-\frac{x+y}
{xy} \)
\(\frac{1}
{y} -\frac{1}
{x}\)
\(\frac{1}
{x} -\frac{1}
{y}\)
9000079208
Level:
B
Assuming \(x\neq 0\)
and \(y\neq 0\),
simplify the following expression.
\[
\left (\frac{x^{-2}y^{2}}
{x^{0}y^{-8}}\right )^{-2} : \frac{x^{2}}
{x^{-4}y^{7}}
\]
\(\frac{1}
{x^{2}y^{13}} \)
\(\frac{y^{13}}
{x^{2}} \)
\(\frac{y^{15}}
{x^{6}} \)
\(\frac{x^{4}}
{y^{27}} \)