Algebraic Expressions, Equations and Identities

The expansion of $a^3$ is ___________ .

Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then the value of $P\left(1\right)$ is

If $x^2-x \cos \left(A+B\right)+1$ is a factor of the expression, $2x^4+4x^3\sin A \sin B-x^2\left(\cos 2A+\cos 2B\right)+4 x \cos A \cos B-2$. Then find the other factor.

Let $P\left(x\right)=a_0+a_{1}x+a_2{x}^2+\dots +a_n{x}^n$ be a polynomial in which $a_i$ is a non-negative integer for each $i\in \left(0,1,2,3,\dots ,n\right)$. If $P\left(1\right)=4$ and $P\left(5\right)=136$, then $P\left(3\right)$ is

How many positive real numbers $x$ are there such that ${\left(x\right)}^{x\sqrt{x}}={\left(x\sqrt{x}\right)}^x$?

Let $r\left(x\right)$ be the remainder when the polynomial $x^{135}+x^{125}-x^{115}+x^5+1$ is divided by $x^3-x.$ Then

Let $P\left(x\right)=x^6-x^5-x^3-x^2-x$ $\&$ $Q\left(x\right)=x^4-x^3-x^2-1$. If $\alpha$, $\beta$, $\gamma$ and $\delta$ are the roots of $Q\left(x\right)=0$ then the value of $P\left(\alpha \right)+P\left(\beta \right)+P\left(\gamma \right)+P\left(\delta \right)$ is equal to

If $f\left(x\right)$ is a polynomial of degree four with the leading coefficient one satisfying $f\left(1\right)=1$, $f\left(2\right)=2$ and $f\left(3\right)=3$, then $\left[\frac{f\left(-1\right)+f\left(5\right)}{f\left(0\right)+f\left(4\right)}\right]$ (where $\left[·\right]$ represents the greatest integer function) is equal to

If $m, n$ are positive integers and $m+n\sqrt{2}=\sqrt{41+24\sqrt{2}}$, then $\left(m+n\right)$ is equal to

$\left(a^2-3a+2\right)x^2+\left(a^2-5a+6\right)x+a-2=r$ for three distinct values of $x$ for some $r\in R, \mathrm{i}\mathrm{f}$ $a+r$ is equal to

Let the sum of the coefficients of the polynomial is $S.$ Find $\frac{S}{16}$
${\left(4x^2-4x+3\right)}^4{\left(4+3x-3x^2\right)}^2$

For natural numbers $x$ and$y,$ let $(x,y)$ denote the greatest common divisor of$x$ and $y .$ How many pairs of natural numbers $x$ and $y$ with $x\le y$ satisfy the equation $xy=x+y+\left(x,y\right)$

The perimeter of picture is $400 \mathrm{cm}$. The length of the picture is $x \mathrm{cm}$. Find the height in terms of $x$.

Expand $(x+1)(x-2)$

If $\alpha$ and $\beta$ are the roots of the equation $x\left(2x-1\right)=0$, find the value of ${\alpha }^2-{\beta }^2$.

If prove that ${\left(x^2-yz\right)}^3+{\left(y^2-zx\right)}^3+{\left(z^2-xy\right)}^3-3\left(x^2-yz\right)\left(y^2-zx\right)\left(z^2-xy\right)$ is a perfect square and hence find its square roots.

If $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x^2}{a}+\frac{y^2}{b}=\frac{ab}{a+b}(a, b\ne 0)$ then prove that $\frac{x^{n+1}}{a}+\frac{y^{n+1}}{b}={\left(\frac{ab}{a+b}\right)}^n$.

Solve for $x$ and $y$ : $x+y+{\left(\frac{x+y}{x-y}\right)}^{\frac{1}{2}}=\frac{12}{x-y}, xy=15$ (given $x>y$)

Solve: $(x-2{)}^{\frac{1}{4}}+(4-x{)}^{\frac{1}{4}}=2$.

Solve for $x$ :

$3x^3=\left\{x^2+\sqrt{18}x+\sqrt{32}\right\}\left(x^2-\sqrt{18}x\right.-\sqrt{32}\}-4x^2$

[Hints: From the given equation try to form $x^4-2{\left(3x+4\right)}^2-x^2\left(3x+4\right)=0$ Then put $3x+4=y$.]