# Test: Algebraic expressions II - Challenging

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Question 1:   Divide ${x}^{3}-3{x}^{2}+4x-2$ by $x-1$
${x}^{2}-2x+2$
${x}^{2}-x$
${x}^{2}-x+2$
${x}^{2}+2x+2$
Question 2:   Factorise: ${x}^{3}-11{x}^{2}+38x-40$
$\left(x-2\right)\left(x-4\right)\left(x-5\right)$
$\left(x-2\right)\left(x-3\right)\left(x-5\right)$
$\left(x-2\right)\left({x}^{2}-9x+20\right)$
$\left(x-2\right){\left(x-5\right)}^{2}$
Question 3:   Simplify $1-\frac{m}{1+\frac{1}{1-\frac{1}{m}}}$ and find value when $m=3$
$-\frac{1}{5}$
$0$
$\frac{1}{5}$
$1$
Question 4:   Simplify expression and find its domain: $\sqrt{{x}^{2}-4x+4}-\sqrt{{x}^{2}}$
$\left\{\begin{array}{l}2\text{ },x\in \left(-\infty ;0\right)\\ -2x+2\text{ },x\in \left[0;2\right)\\ -2\text{ },x\in \left[2;+\infty \right)\end{array}$
$-2$
$-2x+2$
$2$
Question 5:   Divide ${x}^{3}-11{x}^{2}+33x-20$ by ${x}^{2}-7x+5$
$x-5$
$x-3$
$x-4$
${x}^{2}-2x+5$
Question 6:   Find value of expression $\frac{9{y}^{2}+{x}^{2}}{{x}^{2}-xy}$ if $\frac{x}{y}=3$
$2$
$1$
$3$
$0$
Question 7:   Factorise: $\left({x}^{2}+x+1\right)\left({x}^{2}+x+3\right)-20$
$\left({x}^{2}+x+7\right)\left(x-1\right)$
$\left({x}^{2}+x+7\right)\left(x+2\right)\left(x-1\right)$
$\left({x}^{2}+x+7\right)\left(x+2\right)$
$\left({x}^{2}+x+7\right)\left({x}^{2}+x-2\right)$
Question 8:   Simplify expression$\left(\frac{y}{16x-{x}^{3}}-\frac{1}{{x}^{2}+4x}+\frac{4}{{x}^{2}y-16y}\right)÷\frac{{y}^{2}-8y+16}{{x}^{3}y-16xy}$ and find value if $x=0,y=5$
$-2$
$0$
$1$
$-5$
Question 9:   Simplify: $\left({x}^{2}-{y}^{2}\right)÷\left({x}^{-1}-{y}^{-1}\right)$
$\frac{\left(x+y\right)}{xy}$
$-xy\left(x+y\right)$
$xy\left(x+y\right)$
$\frac{xy}{\left(x+y\right)}$
Question 10:   Simplify: $\left(x+\frac{1}{3}\right){\left(3+\frac{1}{x}\right)}^{-1}$
$1$
$3x$
$\frac{3}{x}$
$\frac{x}{3}$