9000083608
Level:
B
Assuming \(xy\neq - 1\),
simplify the expression:
\[
\frac{ \frac{x-y}
{1+xy} + y}
{1 -\frac{y(x-y)}
{1+xy} }
\]
\(x\)
\(\frac{x(1+y^{2})}
{1-y^{2}} \)
\(x - 1\)
\(x(1 + y^{2})\)
Polynomials and Rational Expressions
9000083610
Level:
B
Assuming \(x\neq \pm y\)
and \(y\neq 2x\),
simplify the expression:
\[
\left ( \frac{2x}
{x + y} + \frac{y}
{x - y} - \frac{y^{2}}
{x^{2} - y^{2}}\right ) : \left ( \frac{1}
{x + y} + \frac{x}
{x^{2} - y^{2}}\right )
\]
\(x\)
\(2x - y\)
\(\frac{x}
{2x-y}\)
\(1\)
9000083601
Level:
B
Find the conditions on \(x\)
under which the expression
\[
\frac{\frac{x-y}
{x+y} -\frac{x+y}
{x-y}}
{ \frac{xy}
{x^{2}-y^{2}} }
\]
is well-defined.
\(x\neq 0,\, y\neq 0,\, x\neq \pm y\)
\(x\neq - y\)
\(x\neq \pm y\)
\(x\neq 0,\, y\neq 0\)
9000083605
Level:
A
Find the common denominator of the fractions.
\[
\text{$ \frac{3x}
{x^{2}+4x+4}$ and $ \frac{x+5}
{x^{2}-4}$}
\]
\((x + 2)^{2}(x - 2),\, x\neq \pm 2\)
\((x + 2)(x - 4),\, x\neq \pm 2\)
\((x + 2)^{2}(x - 4),\, x\neq \pm 2\)
\((x + 2)(x - 4),\, x\neq \pm 2\)
9000079206
Level:
A
Assuming \(x\neq 0\),
\(y\neq 0\),
\(x\neq y\),
simplify the following expression.
\[ {
\frac{1}
{x^{2}} - \frac{1}
{y^{2}}
\over -\frac{1}
{y} + \frac{1}
{x}}
\]
\(\frac{x+y}
{xy} \)
\(-\frac{x+y}
{xy} \)
\(\frac{1}
{y} -\frac{1}
{x}\)
\(\frac{1}
{x} -\frac{1}
{y}\)
9000079202
Level:
B
Find the set \(M\)
of all the real \(x\)
for which the following expression is not a well defined number.
\[
\frac{x - 4}
{x^{3} - 16x}
\]
\(M = \{ - 4,0,4\}\)
\(M = \{ - 4,4\}\)
\(M = \{0,4\}\)
\(M = \{0\}\)
9000079204
Level:
B
Find the domain of the following expression.
\[
\frac{x^{2} - x}
{x + 1} : \frac{x^{2} - 1}
{x^{2} + 2x + 1}
\]
\(\mathbb{R}\setminus \{ - 1,1\}\)
\(\mathbb{R}\setminus \{ - 1,0,1\}\)
\(\mathbb{R}\setminus \{ - 1\}\)
\(\mathbb{R}\setminus \{ - 1,0\}\)
9000079208
Level:
B
Assuming \(x\neq 0\)
and \(y\neq 0\),
simplify the following expression.
\[
\left (\frac{x^{-2}y^{2}}
{x^{0}y^{-8}}\right )^{-2} : \frac{x^{2}}
{x^{-4}y^{7}}
\]
\(\frac{1}
{x^{2}y^{13}} \)
\(\frac{y^{13}}
{x^{2}} \)
\(\frac{y^{15}}
{x^{6}} \)
\(\frac{x^{4}}
{y^{27}} \)
9000079205
Level:
A
Assuming \(x\neq 0\)
and \(x\neq 2\),
simplify the following expression.
\[
\frac{x^{3} - x^{2}}
{x - 2} \cdot \frac{2 - x}
{x^{2}}
\]
\(1 - x\)
\(x - 1\)
\(x + 1\)
\(x^{2} - 1\)
9000079201
Level:
A
Evaluate
\[
\frac{-x^{2}}
{x - y} -\frac{y - x}
{x + y}
\]
at \(x = -1\),
\(y = 2\).
\(-\frac{8}
{3}\)
\(-\frac{10}
{3} \)
\(-\frac{2}
{3}\)
\(-\frac{4}
{3}\)