Factorise:$x^6-a^6$

Important Questions on Polynomials

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

Let $x={\left(\sqrt{50}+7\right)}^{\frac{1}{3}}-{\left(\sqrt{50}-7\right)}^{\frac{1}{3}}$. Then-

EASY

Mathematics>Algebra>Polynomials>Algebraic Identities

The coefficient of $x$ in ${\left(x-3y\right)}^3$ is:

HARD

Mathematics>Algebra>Polynomials>Algebraic Identities

If $a^3+\frac{1}{a^3}=52$, then the value of $2\left(a+\frac{1}{a}\right)$ is:

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

Let $a,b,c$ be non-zero real numbers such that $a+b+c=0;$ let $q=a^2+b^2+c^2 \& r=a^4+b^4+c^4$. Then

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If $x-y=4$ and $xy=45$, then the value of $x^3-y^3$ is:

HARD

Mathematics>Algebra>Polynomials>Algebraic Identities

If $a^4+\frac{1}{a^4}=50, a>0,$ then find the value of $a^3+\frac{1}{a^3}$.

EASY

Mathematics>Algebra>Polynomials>Algebraic Identities

If $a+\frac{1}{a}=5$ then $a^3+\frac{1}{a^3}$ is:

EASY

Mathematics>Algebra>Polynomials>Algebraic Identities

${(3a-4b)}^3$ is equal to:

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If $\mathrm{p}+\mathrm{q}=7$ and $\mathrm{pq}=5$, then the value of ${\mathrm{p}}^3+{\mathrm{q}}^3$is:

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If P$=\frac{x^4-8x}{x^3-x^2-2x}$, Q$=\frac{x^2+2x+1}{x^2-4x-5}$ and R$=\frac{2x^2+4x+8}{x-5}$, then $(P\times Q)÷R$ is equal to:

EASY

Mathematics>Algebra>Polynomials>Algebraic Identities

If ${\mathrm{a}}^2+{\mathrm{b}}^2-{\mathrm{c}}^2=0$, then the value of $\frac{2\left({\mathrm{a}}^6+{\mathrm{b}}^6-{\mathrm{c}}^6\right)}{3{\mathrm{a}}^2{\mathrm{b}}^2{\mathrm{c}}^2}$ is:

HARD

Mathematics>Algebra>Polynomials>Algebraic Identities

Consider the equation ${\left(1+a+b\right)}^2=3\left(1+a^2+b^2\right),$ where $a, b$ are real numbers. Then

EASY

Mathematics>Algebra>Polynomials>Algebraic Identities

Expand: ${(4a+3b+2c)}^2$

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If $a,b,c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$ , then ${\left(3a+5b-8c\right)}^2+{\left(-8a+3b+5c\right)}^2+{\left(5a-8b+3c\right)}^2$ is equal to

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If $A+B=12$ and $AB=17$, what is the value of $A^3+B^3$?

EASY

Mathematics>Algebra>Polynomials>Algebraic Identities

If ${(2x–5y)}^3 – {(2x+ 5y)}^3 = y\left(A{x}^2+B{y}^2\right)$, then what is the value of $\left(2A-B\right)$?

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If ${\mathrm{a}}^3+{\mathrm{b}}^3=416$ and $\mathrm{a}+\mathrm{b} =16$, then ${\left(\mathrm{a}-\mathrm{b}\right)}^2+\mathrm{ab}$ is equals to_____.

HARD

Mathematics>Algebra>Polynomials>Algebraic Identities

If ${\mathrm{x}}^2-2\sqrt{5}\mathrm{x}+1=0$, then what is the value of  ${\mathrm{x}}^5+{\mathrm{x}}^{\frac{1}{5}}$:

MEDIUM

Mathematics>Algebra>Polynomials>Algebraic Identities

If the difference of two numbers is $5$ and the difference of their square is $125$, then out of the two which is the smaller number?

HARD

Mathematics>Algebra>Polynomials>Algebraic Identities

Let $x$ and $y$ be two $2$-digit numbers such that $y$ is obtained by reversing the digits of $x.$ Suppose they also satisfy $x^2-y^2=m^2$ for some positive integer $m.$ The value of $x+y+m$ is.