Question

Question 14
If (x + 1) is a factor of f(x) = $2x^3+a{x}^2+2bx+1$, then find the value of a and b is given that 2a – 3b = 4.

Answer 1

Given that, (x + 1) is a factor f(x) = $2x^5+a{x}^2+2bx+1$ then f(-1) = 0
[if (x + $\alpha$) is a factor of f(x) = $a{x}^2$ + bx + c, then f(-) = 0]
$\Rightarrow 2(-1{)}^3+a(-1{)}^2+2b(-1)+1=0$
$\Rightarrow$ -2 + a – 2b + 1 = 0
$\Rightarrow$ a – 2b – 1 = 0
$\Rightarrow$ 2a – 3b = 4
$\Rightarrow$ 3b = 2a – 4
$\Rightarrow b=\left(\frac{2a-4}{3}\right)$
Now, put the value of b in Eq. (i), we get
$a-2\left(\frac{2a-4}{3}\right)-1=0$
$\Rightarrow$ 3a – 2(2a – 4) – 3 = 0
$\Rightarrow$ 3a – 4a + 8 – 3 = 0
$\Rightarrow$ - a + 5 = 0
$\Rightarrow$ a = 5
Now, put the value of b in Eq. (i), we get
$a-2\frac{2a-4}{3}-1=0$
$\Rightarrow$ 3a – 2(2a – 4) – 3 = 0
$\Rightarrow$ - a + 5 = 0
$\Rightarrow$ a = 5
Now, put the value of a in Eq (i), we get
5 – 2b – 1 = 0
$\Rightarrow$ 2b = 4
$\Rightarrow$ b = 2
Hence, the required values of a and b are 5 and 2, respectively