Find the values of “a” and “b” so that (x + 2) and (x – 1) may be factors of $x^3+10x^2+ax+bx^3\; +\; 10x^2\; +\; ax\; +\; b$

Simplify $\frac{(x+y{)}^2-z^2}{(x+y+z{)}^2}÷\{\frac{(x-z{)}^2-y^2}{x^2+xy+zx}÷\frac{(x-y{)}^2-z^2}{x^2-xy+zx}\}\mathrm{.}\backslash frac\{(x+y)^2-z^2\}\{(x+y+z)^2\}\backslash div\backslash left\backslash \{\backslash frac\{(x-z)^2-y^2\}\{x^2+xy+zx\}\backslash div\backslash frac\{(x-y)^2-z^2\}\{x^2-xy+zx\}\backslash right\backslash \}.$

The expression $2x^3+a{x}^2+bx\text{–}22x^3\; +\; ax^2\; +\; bx\; –\; 2$ leaves a remainder 7 and 0 when divided by $(2x-3)(2x-3)$ and $(x+2)(x+2)$ Find the values of a and b.

Find the value of k if the expressions $p(x)=k{x}^3+4x^2+3x\text{–}4p(x)\; =\; kx^3\; +\; 4x^2\; +\; 3x\; –\; 4$ and $q(x)=x^3\text{–}4x+kq(x)\; =\; x^3\; –\; 4x\; +\; k$ leave the same remainder when divided by $(x-3)(x-3)$

If a + c + e = 0 and b + d = 0 then $a{x}^4+b{x}^3+c{x}^2+dx+eax^4\; +\; bx^3\; +cx^2\; +\; dx\; +\; e$ is exactly divisible by ..................

Simplify $\frac{1}{1+x+x^2}-\frac{1}{1-x+x^2}+\frac{2x}{1+x^2+x^4}\backslash frac\{1\}\{1+x+x^2\}-\backslash frac\{1\}\{1-x+x^2\}+\backslash frac\{2x\}\{1+x^2+x^4\}$ ...............

What should be subtracted from $x^3\text{–}2x^2+4x+1x^3\; –\; 2x^2\; +\; 4x\; +\; 1$

What should be subtracted from $\frac{7x}{x^2-x-12}\backslash frac\{7x\}\{x^2-x-12\}$ to get $\frac{3}{x+3}\backslash frac\{3\}\{x+3\}$ ?

If $a{x}^2+2a^{2}x+b^{3}ax^2\; +\; 2a^2\; x\; +\; b^3$ is divisible by (x + a) then ……………

Which of the following expressions is a rational expression ?