The polynomial equation of degree 5 whose roo | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the roots of the given equation $x^5-3x^4-x^3+11x^2-12x+4=0$ be $\alpha_i$. We want the equation whose roots are $\beta_i = \alpha_i + 2$. Thi

Vedclass · Accepted Answer

$x^5-13x^4+63x^3-135x^2+108x=0$

Vedclass · Answer

(C) Let the roots of the given equation $x^5-3x^4-x^3+11x^2-12x+4=0$ be $\alpha_i$. We want the equation whose roots are $\beta_i = \alpha_i + 2$. This implies $\alpha_i = \beta_i - 2$. Substituting $x = y - 2$ into the original equation,we get $(y-2)^5 - 3(y-2)^4 - (y-2)^3 + 11(y-2)^2 - 12(y-2) + 4 = 0$. Expanding each term: $(y-2)^5 = y^5 - 10y^4 + 40y^3 - 80y^2 + 80y - 32$ $-3(y-2)^4 = -3(y^4 - 8y^3 + 24y^2 - 32y + 16) = -3y^4 + 24y^3 - 72y^2 + 96y - 48$ $-(y-2)^3 = -(y^3 - 6y^2 + 12y - 8) = -y^3 + 6y^2 - 12y + 8$ $11(y-2)^2 = 11(y^2 - 4y + 4) = 11y^2 - 44y + 44$ $-12(y-2) = -12y + 24$ $+4 = 4$ Summing these: $y^5 + (-10-3)y^4 + (40+24-1)y^3 + (-80-72+6+11)y^2 + (80+96-12-44-12)y + (-32-48+8+44+24+4) = 0$ $y^5 - 13y^4 + 63y^3 - 135y^2 + 108y = 0$. Thus,the equation is $x^5 - 13x^4 + 63x^3 - 135x^2 + 108x = 0$.

The polynomial equation of degree $5$ whose roots are the roots of the equation $x^5-3x^4-x^3+11x^2-12x+4=0$ each increased by $2$,is

Similar Questions

The number of real values of $x$ for which the equality $|3x^2 + 12x + 6| = 5x + 16$ holds is:

$A$ two-digit number is four times the sum of its digits and three times the product of its digits. The number is:

The equation $(\cos p - 1) x^2 + (\cos p) x + \sin p = 0$ in the variable $x$ has real roots. Then $p$ can take any value in the interval

The number of solution$(s)$ of the equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ is/are

If the quadratic equation $(\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \lambda \neq -2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module