Question

If /alpha,/beata,/gamma are the zeros of polynomial p(x) = x? + px2 + x + 2 such that aß + 1 =
o then find the value of 2p+q.​

Answer 1

Answer:

$α,β,γ zeroes of polynomial x3+px2+qx2+2 i.e, on keeping x=αorβorγ, we will get x3+px+qx+2=0</p><p></p><p>Also, αβ+1=0</p><p></p><p>We know that sum of roots of cubic polynomial =1−p</p><p></p><p>⇒α+β+γ=−p & αβ+βγ+γα=q</p><p></p><p>⇒αβγ=−2</p><p></p><p>Since αβγ=−2</p><p></p><p>& αβ+1=0⇒αβ=−1</p><p></p><p>⇒(−1)γ=−2</p><p></p><p>⇒γ=2</p><p></p><p>∴α+β+2=−p  ⇒α+β=−p−2→(1)</p><p></p><p>    αβ+βγ+γα=q⇒−1+2β+2α=q</p><p></p><p>⇒(α+β)=2(q+1)→(2)</p><p></p><p>Equating (1)&(2)</p><p></p><p>⇒−p−2=2q+1</p><p></p><p>⇒−2p−4=q+1</p><p></p><p>⇒2p+q+5=0</p><p></p><p>Hence, the answer is 2p+q+5=0.$