If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

If α, β, and γ are the roots of the equation x³ + px² + qx + r = 0, find the value of  \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

Solve the equation 3x³ - 16x² + 23x - 6 = 0 if the product of two roots is 1.

Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

Find the sum of the squares of the roots of ax⁴+ bx³+ cx²+ dx + e = 0. \(a \neq 0\)

Find the condition that the roots of cubic x³+ ax²+ bx + c = 0 are in the ratio p : q : r.

Form the equation whose roots are the squares of the roots of the cubic equation x³+ ax²+ bx + c = 0.

Form a polynomial equation with integer coefficients with \(\sqrt { \frac { \sqrt { 2 } }{ \sqrt { 3 } } } \) as a root.

Prove that a line cannot intersect a circle at more than two points.

If k is real, discuss the nature of the roots of the polynomial equation 2x²+ kx + k = 0, in terms of k.

Solve the equation x⁴-9x²+20 = 0.

Solve the equation x³-3x²- 33x + 35 = 0.

Solve the equation 2x³+11x²−9x−18 = 0.

If the roots of x³+ px²+ qx + r = 0 are in H.P. prove that 9pqr = 27r²+2q³.

Solve the cubic equation : 2x³−x²−18x + 9 = 0 if sum of two of its roots vanishes.

Determine k and solve the equation 2x³-6x²+3x+k = 0 if one of its roots is twice the sum of the other two roots.

Solve the equation
2x³ - 9x² + 10x = 3

Solve the equation x³- 5x²- 4x + 20 = 0

Find the roots of 2x³ + 3x² + 2x + 3 = 0

Solve the equation 7x ³ -  43x ²  =  43x - 7

Find solution, if any, of the equation 2cos²x - 9cosx + 4 = 0

Solve the following equations,
sin²x - 5 sinx + 4 = 0

Examine for the rational roots of 2x³- x²- 1 = 0

Solve: \(8x^{ \frac { 3 }{ 2x } }-8x^{ \frac { -3 }{ 2x } }\) = 63

Show that the polynomial 9x⁹+ 2x⁵- x⁴- 7x²+ 2 has at least six imaginary roots.

Discuss the nature of the roots of the following polynomials:
x²⁰¹⁸+1947x¹⁹⁵⁰+15x⁸+26x⁶+2019

Solve: \(2\sqrt { \frac { x }{ a } } +3\sqrt { \frac { a }{ x } } =\frac { b }{ a } +\frac { 6a }{ b } \)

Find all real numbers satisfying 4^x- 3(2^x+2) + 2⁵ = 0

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x⁹- 4x⁸+ 4x⁷- 3x⁶+ 2x⁵+ x³+7x²+7x+2 = 0

Discuss the maximum possible number of positive and negative roots of the polynomial equations x²−5x+6 and x²−5x+16 . Also draw rough sketch of the graphs

Determine the number of positive and negative roots of the equation x⁹- 5x⁸-14x⁷= 0.

Find the exact number of real zeros and imaginary of the polynomial x⁹+9x⁷+7x⁵+5x³+3x.

Solve the cubic equations: 8x³ - 2x² - 7x + 3 = 0

Solve the following equations,
12x³+ 8x = 29x²- 4

Examine for the rational roots of x⁸- 3x + 1 = 0

Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+ 2x³+ 5x²+11

Every polynomial equation of degree n ≥ 1 has at least one root in C. (The Fundamental Theorem of Algebra)

If a complex number z₀ is a root of a polynomial equation with real coefficients, then its complex conjugate \(\bar{z}_{0}\) is also a root. (Complex Conjugate Root Theorem)

Let p and q be rational numbers such that \(\sqrt{q}\) is irrational. If p + \(\sqrt{q}\) is a root of a quadratic equation with rational coefficients, then p − \(\sqrt{q}\) is also a root of the same equation.

Let p and q be rational numbers so that \(\sqrt{p} \text { and } \sqrt{q}\) are irrational numbers; further let one of \(\sqrt{p} \text { and } \sqrt{q}\) be not a rational multiple of the other. If \(\sqrt{p}+\sqrt{q}\) is a root of a polynomial equation with rational coefficients, then \(\sqrt{p}-\sqrt{q},-\sqrt{p}+\sqrt{q}, \text { and }-\sqrt{p}-\sqrt{q}\) are also roots of the same polynomial equation.

Let \(a_{n} x^{n}+\cdots+a_{1} x+a_{0} \text { with } a_{n} \neq 0 \text { and } a_{0} \neq 0\) be a polynomial with integer coefficients . If \(\frac{p}{q}\) with ( p, q) = 1, is a root of the polynomial, then p is a factor of a0 and q is a factor of a_n. (Rational Root Theorem)

A polynomial equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}=0, \ \left(a_{n} \neq 0\right)\) is a reciprocal equation if, and only if, one of the following two statements is true:
\((i)\ a_{n}=a_{0}, \ a_{n-1}=a_{1}, \ a_{n-2}=a_{2} \cdots \)
\((ii)\ a_{n}=-a_{0}, a_{n-1}=-a_{1}, a_{n-2}=-a_{2}, \cdots\)

If p is the number of positive zeros of a polynomial P(x) with real coefficients and s is the number of sign changes in coefficients of P(x), then s − p is a nonnegative even integer. (Descartes Rule)