9000101606
Level:
B
Expand \(\left (x - y\right )^{3} - x\left (x + y\right )^{2}\).
\(- y^{3} - 5x^{2}y + 2xy^{2}\)
\(y^{3} - 5x^{2}y + 2xy^{2}\)
\(- y^{3} - 5x^{2}y - 4xy^{2}\)
\(- y^{3} - 5x^{2}y + 4xy^{2}\)
Polynomials and fractions
9000101610
Level:
C
Simplify using long division of polynomials.
\[
\left (2x^{3} - x^{2} - 3x - 1\right ) : \left (2x + 1\right )
\]
\(x^{2} - x - 1\)
\(x^{2} - x + 1\)
\(x^{2} + x + 1\)
\(x^{2} - 2x - 1\)
9000101701
Level:
B
Factor the following polynomial.
\[
15xy - 10x - 3y + 2
\]
\(\left (5x - 1\right )\left (3y - 2\right )\)
\(5x\left (3y - 2\right )\)
\(4x\left (3y - 2\right )\)
\(- 5x\left (3y - 2\right )\)
9000101609
Level:
C
Simplify using long division of polynomials.
\[
\left (3x^{3} + 17x^{2} + 23x + 5\right ) : \left (x^{2} + 4x + 1\right )
\]
\(3x + 5\)
\(3x - 5\)
\(3x + 1\)
\(3x - 1\)
9000101705
Level:
B
Factor the following polynomial expression.
\[
16a^{2}b^{2} - 4a^{2}c^{2} - 16b^{2}d^{2} + 4c^{2}d^{2}
\]
\(4\left (a - d\right )\left (a + d\right )\left (2b + c\right )\left (2b - c\right )\)
\(4\left (a + b\right )^{2}\left (2b + c\right )^{2}\)
\(4\left (a - b\right )\left (a + b\right )\left (2b + c\right )\left (2b - c\right )\)
\(4\left (a - c\right )\left (a + c\right )\left (2b + d\right )\left (2b - d\right )\)
9000101603
Level:
A
Simplify the polynomial \((x + 1)(x - 1)^{2} - (x - 1)(x + 1)^{2}\)
into one of the following forms.
\(- 2\left (x - 1\right )\left (x + 1\right )\)
\(2\left (x - 1\right )\left (x + 1\right )\)
\(0\)
\(2\)
9000101707
Level:
C
Factor the following polynomial.
\[
x^{6} - 1
\]
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + x + 1\right )\left (x^{2} - x + 1\right )\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + x + 1\right )\left (x^{2} - x - 1\right )\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + 2x + 1\right )\left (x^{2} - 2x + 1\right )\)
\(\left (x - 1\right )\left (x + 1\right )\left (x^{2} + x - 1\right )\left (x^{2} - x + 1\right )\)
9000088810
Level:
A
Simplify the following expression.
\[
\left (x -\frac{1}
{x}\right )\cdot \left (1 - \frac{x}
{x + 1}\right )
\]
\(\frac{x - 1}
{x} \)
\(\frac{x - 1}
{x + 1}\)
\(\frac{1 - x}
{x + 1}\)
\(\frac{1 - x}
{x} \)
9000087505
Level:
C
Assuming \(x\in \mathbb{R}\setminus \left \{-\frac{1}
{2}\right \}\),
find the quotient of the polynomials.
\[
(4x^{3} - 1) : (2x + 1)
\]
\(2x^{2} - x + \frac{1}
{2} - \frac{\frac{3}
{2} }
{2x+1}\)
\(2x^{2} + x + \frac{1}
{2} - \frac{\frac{3}
{2} }
{2x+1}\)
\(2x^{2} - x + \frac{1}
{4} - \frac{\frac{3}
{2} }
{2x+1}\)
\(2x^{2} + x + \frac{1}
{4} - \frac{\frac{3}
{2} }
{2x+1}\)
9000087506
Level:
C
Assuming \(x\in \mathbb{R}\setminus \left \{1\right \}\),
find the quotient of the polynomials.
\[
(2x + 2x^{2} - 3) : (x - 1)
\]
\(2x + 4 + \frac{1}
{x-1}\)
\(2x + 4 + \frac{2}
{x-1}\)
\(2x + 2 + \frac{1}
{x-1}\)
\(2x + 2 + \frac{2}
{x-1}\)