1003032401
Level:
B
Factoring the polynomial \( 625x^4-1 \) we get:
\( \left( 25x^2+1\right)(5x-1)(5x+1) \)
\( (5x-1)(5x+1)^2 \)
\( (5x-1)^4 \)
\( \left( 25x^2+1\right)(5x-1)^2 \)
Polynomials and fractions
1003032501
Level:
B
The product \( (x+y)\left(x^2+y^2\right)\left(x^3+y^3\right) \) equals:
\( x^6+x^5y+x^4y^2+2x^3y^3+x^2y^4+xy^5+y^6 \)
\( x^6-x^5y+x^4y^2-2x^3y^3+x^2y^4-xy^5+y^6 \)
\( (x+y)^6 \)
\( x^6+y^6 \)
1003032504
Level:
B
Factoring the polynomial \( -4x^4+26x^3-12x^2 \) you get:
\( -2x^2(x-6)(2x-1) \)
\( 2x^2(x+1)(2x-1) \)
\( -2x^2(x+6)(2x+1) \)
\( 2x^2(x-6)(2x+1) \)
2010000814
Level:
B
Assuming \(x\neq 0\), \(y\neq 0\), \(y\neq \pm 1\), simplify the expression:
\[\left [\left ( \frac{y-1} {y}\right )^{2} : \left (\frac{x} {y+1} \right )^{2}\right ] : \frac{2(y^2-1)} {xy}\]
\(\frac{y^2-1}
{2xy}\)
\( 2\)
\(\frac{y^2-1}
{2}\)
\(\frac{y-1}
{2}\)
2010000901
Level:
B
Assuming \(xy\neq 1\),
simplify the expression:
\[
\frac{ \frac{x+y}
{1-xy} -x}
{1 +\frac{x(x+y)}
{1-xy} }
\]
\( y\)
\(\frac{y(1+x^{2})}
{1-x^{2}} \)
\(\frac{y}
{1+x^{2}} \)
\( y(1+x^2)\)
2010000902
Level:
B
Assuming \(x\neq \pm y\)
and \(x\neq 0\),
simplify the expression:
\[
\left ( \frac{y}
{y-x} - \frac{2x}
{y+x} - \frac{y^{2}}
{y^{2} - x^{2}}\right ) : \left ( \frac{1}
{x + y} - \frac{y}
{y^{2} - x^{2}}\right )
\]
\( y-2x\)
\(2x-y\)
\(\frac{2x-y}
{x}\)
\( 0\)
2010000905
Level:
B
Suppose we are given the following equality of two fractions with nonzero denominators. From the given expressions, choose the one that by substituting to the starred position makes the equality true.
\[
\frac{2- 3x}
{x +2} = \frac{2(9x^{2} - 12x + 4)}
{*}
\]
\((2x +4)(2 - 3x)\)
\((x +2)(2 - 3x)\)
\((x +2)(4 - 9x)\)
\((2x +4)(3x - 2)\)
2010001303
Level:
B
Expand \(\left (x + y\right )^{3} - y\left (x -y\right )^{2}\).
\(x^{3} + 2x^{2}y + 5xy^{2}\)
\(x^{3} + 2x^{2}y + xy^{2}\)
\(x^{3} + 2x^{2}y + xy^{2}+2y^3\)
\(x^{3} + 2x^{2}y + 5xy^{2}+2y^3\)
2010001304
Level:
B
Factor the following polynomial.
\[
20xy + 12y - 5x - 3
\]
\(\left (4y - 1\right )\left (5x +3\right )\)
\(4y\left (5x +3\right )\)
\(\left (1-4y\right )\left (5x +3\right )\)
\(- 4y\left (5x +3\right )\)
2010001305
Level:
B
Factor the following polynomial expression.
\[
36b^{2}c^{2} - 9a^{2}b^{2} - 36c^{2}d^{2} + 9a^{2}d^{2}
\]
\(9\left (b - d\right )\left (b + d\right )\left (2c + a\right )\left (2c - a\right )\)
\(\left (b^2 + d^2\right )\left (36c^2 + 9a^2\right )\)
\(9\left (a - d\right )\left (a + d\right )\left (2b + c\right )\left (2b - c\right )\)
\(\left (a^2 + d^2\right )\left (36b^2 + 9c^2\right )\)