Courses
Courses for Kids
Free study material
Free LIVE classes
More LIVE
Join Vedantu’s FREE Mastercalss

# If $x+1$ is a factor of $2{{x}^{3}}+a{{x}^{2}}+2bx+1$ ,then find the value of a and b given that $2a-3b=4$ Verified
334.8k+ views
Hint: Let there be any cubic equation as $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ which has three roots as $\alpha ,\beta ,\gamma$
So we can write this cubic equation as $a(x-\alpha )(x-\beta )(x-\gamma )=0$, Now we can also say that $(x-\alpha ),(x-\beta )and(x-\gamma )$ are the factors of cubic equation $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ . We will obtain an equation in terms of a and b substituting factor which is x+1=0 or x=-1 in the given equation. We have already been given another equation in a and b, so now using elimination method, we can compute values of a and b.

In the question we are given a cubic equation as $2{{x}^{3}}+a{{x}^{2}}+2bx+1=0$ which has a factor $x+1$
Assuming the roots are $\alpha ,\beta ,\gamma$ so we can write our cubic equation as
$2(x-\alpha )(x-\beta )(x-\gamma )=0$
And factors of this equation will be $(x-\alpha ),(x-\beta )and(x-\gamma )$
But one factor given as $x+1$
So, replacing $(x-\alpha )$ with $x+1$
Now our equation can be written as $2(x+1)(x-\beta )(x-\gamma )=0$
Looking this carefully if we make $(x+1)=0$ then this cubic equation will become 0
It means $(x+1)=0$ gives $x=-1$ is a solution of this cubic equation
putting $x=-1$ in equation $2{{x}^{3}}+a{{x}^{2}}+2bx+1$ will make this equation 0
$2{{(-1)}^{3}}+a{{(-1)}^{2}}+2b(-1)+1=0$
$\to (-2+a-2b+1)=0$
On solving gives
$a-2b=1.......(1)$
Now $a$and $b$are unknown
But in the question, we are given an equation $2a-3b=4.......(2)$
So, we have 2 equation and 2 variables $a$ and $b$
We can calculate them but elimination method
So first we eliminate $a$
Multiplying the equation (1) with 2
Now equation (1) will look like
$2a-4b=2.........(3)$
On subtracting equation (3) from equation (2), we get
$2a-3b-(2a-4b)=4-2$
$2a-3b-2a+4b=2$
$b=2$
Now we get $b=2$ so we can put this value in equation (1) to get the value of $a$
Equation (1) $a-2b=1$
$a-2(2)=1$
$a=5$
Hence, we get the value $a=5$ and $b=2$

Note: In this question students can make mistakes, by taking its root as 1 and not -1. This question is very basic; you just need to solve equations carefully. You can also calculate the values of $a$ and $b$ by substitution method.
Like we are given these two equations so we find value of $a$ from equation (1) and put it in equation (2)
$a-2b=1.......(1)$ $2a-3b=4.......(2)$
$a=1+2b$ from equation (1)
Putting value of $a$ into equation (2)
We get
$2(1+2b)-3b=4$
$\to 2+4b-3b=4$
$\to 2+4b-3b=4$
$\to b=2$
And we have $a=1+2b$ so putting value $b=2$
$\to a=5$
Last updated date: 24th Sep 2023
Total views: 334.8k
Views today: 3.34k