9000088809
Level:
A
Simplify the following expression.
\[
\left ( \frac{1}
{m - n} - \frac{1}
{m + n}\right )\cdot \left (\frac{m^{2} + 2mn + n^{2}}
{2n} \right )
\]
\(\frac{m+n}
{m-n}\)
\(0\)
\(\frac{m(m+n)}
{n(m-n)} \)
\(2\)
Polynomials and Rational Expressions
9000088808
Level:
B
Find the common denominator of the following three fractions.
\[
\text{$ \frac{a}
{a^{2}-ab}\ ,\qquad \frac{-b}
{a^{2}-b^{2}} \ ,\qquad \frac{2b}
{ab+b^{2}} $}
\]
\(ab(a^{2} - b^{2})\)
\(ab(a - b)\)
\(ab(a + b)\)
\(ab(a + b)^{2}\)
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:
\[
(x^{2} + x + 1) : (2x + 3)
\]
\(\frac{1}
{2}x -\frac{1}
{4} + \frac{\frac{7}
{4} }
{2x+3}\)
\(\frac{1}
{2}x -\frac{1}
{2} + \frac{\frac{7}
{4} }
{2x+3}\)
\(x + 2 + \frac{7}
{2x+3}\)
\(x - 2 + \frac{7}
{2x+3}\)
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.
\(x\neq 0,\; y\neq 0,\; x\neq \pm y\)
\(x\neq - y\)
\(x\neq \pm y\)
\(x\neq 0,\; y\neq 0\)
9000083602
Level:
A
Evaluate the following expression at \(x = \frac{1}
{2}\).
\[
\frac{x^{2} - 2}
{1 -\frac{1}
{x}}
\]
\(\frac{7}
{4}\)
\(-\frac{7}
{4}\)
\(\frac{7}
{2}\)
\(-\frac{7}
{2}\)
9000083605
Level:
A
Find the common denominator of the fractions.
\[
\text{$ \frac{3x}
{x^{2}+4x+4}$ and $ \frac{x+5}
{x^{2}-4}$}
\]
\((x + 2)^{2}(x - 2),\; x\neq \pm 2\)
\((x + 2)(x - 4),\; x\neq \pm 2\)
\((x + 2)^{2}(x - 4),\; x\neq \pm 2\)
\((x + 2)(x - 4),\; x\neq \pm 2\)
9000083609
Level:
B
Assuming \(x\neq 0\),
\(x\neq \pm y\),
\(y\neq 0\),
simplify the expression.
\[
\frac{\frac{x^{2}+y^{2}}
{x} - 2y}
{\left ( \frac{1}
{y^{2}} - \frac{1}
{x^{2}} \right )\cdot \frac{xy}
{x+y}}
\]
\(y(x - y)\)
\(\frac{x-y}
{y} \)
\(x(x - y)\)
\(\frac{x-y}
{x} \)
9000083603
Level:
A
Evaluate the following expression at \(x = \frac{1}
{2}\)
and \(y = -\frac{1}
{4}\).
\[
\frac{x -\frac{y}
{x}}
{1 + \frac{x}
{y}}
\]
\(- 1\)
\(3\)
\(4\)
\(1\)
9000083604
Level:
A
Assuming \(x\neq - 1\),
\(x\neq \pm y\),
simplify the expression.
\[
\frac{x^{2} + 2xy + y^{2}}
{2x^{2} + 4x + 2} \cdot \frac{(x + 1)(y - x)}
{y^{2} - x^{2}}
\]
\(\frac{x+y}
{2x+2}\)
\(\frac{x+y}
{2} \)
\(x + y\)
\(\frac{1}
{2}\)