CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Given that the zeros of the cubic polynomial $${x}^{3}-6{x}^{2}+3x+10$$ are of the form $$a,a+b$$ and $$a+2b$$ for some real numbers $$a$$ and $$b$$, find the values of $$a$$ and $$b$$


Solution

Given that $$a,a+b\ \&a+2b$$ are roots of given polynomila $$x^3-6x^2+3x+10$$
Sum of the roots $$=a+2b+a+a+b=\dfrac{-coefficient\ of\ x^2}{coefficient\ of\ x^3}$$
$$\Rightarrow 3a+3b=\dfrac{-(-6)}{1}=6$$
$$\Rightarrow a+b=2.......(1)$$  $$b=2-a$$
Product of roots $$=(a+2b)(a+b)a=\dfrac{-constant}{coefficient\ of\ x^3}$$
$$\Rightarrow (a+b+b)(a+b)a=\dfrac{-10}{1}=-10$$
$$\Rightarrow (2+b)(2)a=-10$$
$$\Rightarrow (2+2-a)2a=-10$$
$$\Rightarrow (4-a)2a=-10$$
$$\Rightarrow 4a-a^2=-5$$
$$\Rightarrow a^2-4a-5=0$$
$$\Rightarrow a^2-5a+a-5=0$$
$$\Rightarrow (a-5)(a+1)=0$$
$$a=5\ or\ a=-1$$
when $$a=5,\ 5+b=2\Rightarrow b=-3$$
          $$a=-1,\ -1+b=2\Rightarrow b=3$$

Mathematics

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image