Please answer these Questions - What real number is to be subtracted from 3x3+10x2-14x+9 so that the polynomial x-1 - Maths -

Question

Please answer these Questions -

What real number is to be subtracted from
3x3+10x2-14x+9 so that the polynomial x-1
divides it exactly
Find the condition which must be satisfied by the coefficient
of polynomial x3-px2+qx-r, when sum of its two zeroes
are 0
If product of the two zeroes is 4 of the polynomials 1
ax2-6x-6

2 (a2+9)x2+13x+6a
Find 'a'
4 If the zeroes of the given Polynomial 4x2-2x+k-4
are reciprocal to each other, then find k

Koka Sri Lakshmi Divya Sai · Accepted Answer

$Given 3 x^{3} + 10 x^{2} - 14 x + 9 x - 1 divides the polynomial$

8 must be subtracted from the polynomial 2 $Let the zeroes be a-d,a,a+d \Rightarrow a - d + a + a + d = p \Rightarrow 3 a = p \Rightarrow a = \frac{p}{3} So a is the zero of the polynomial \Rightarrow a^{3} - p a^{2} + q a - r = 0 \Rightarrow (\frac{p}{3})^{3} - p (\frac{p}{3})^{2} + q (\frac{p}{3}) - r = 0 \Rightarrow \frac{p^{3}}{27} - \frac{p^{3}}{9} + \frac{p q}{3} - r = 0 \Rightarrow p^{3} - 3 p^{3} + 9 p q - 27 r = 0 \Rightarrow - 2 p^{3} + 9 p q - 27 r = 0 \Rightarrow 2 p^{3} - 9 p q + 27 r = 0 3 (i) Given {ax}^{2} - 6 x - 6 = 0 Let α, β are the zeroes of the polynomial \Rightarrow α β = \frac{- 6}{a} \Rightarrow 4 = \frac{- 6}{a} \Rightarrow a = - \frac{3}{2} (i i) {(a}^{2} + 9) x^{2} + 13 x + 6 a Data is missing 4 Given 4 x^{2} - 2 x + (k - 4) = 0 Let α and \frac{1}{α} be the zeroes of the polynomial \Rightarrow α \frac{1}{α} = \frac{k - 4}{4} \Rightarrow 1 = \frac{k - 4}{4} \Rightarrow 4 = k - 4 \Rightarrow k = 8$