Find the Values of a and b

Video Explanation

Question

If \(x + 1\) is a factor of \[ 2x^3 + ax^2 + 2bx + 1, \] find the values of \(a\) and \(b\) given that \[ 2a – 3b = 4. \]

Solution

Step 1: Use Factor Theorem

Since \(x + 1\) is a factor, substitute \(x = -1\):

\[ 2(-1)^3 + a(-1)^2 + 2b(-1) + 1 = 0 \]

\[ -2 + a – 2b + 1 = 0 \]

\[ a – 2b – 1 = 0 \]

\[ a – 2b = 1 \quad (1) \]

Step 2: Use the Given Equation

\[ 2a – 3b = 4 \quad (2) \]

Step 3: Solve Equations (1) and (2)

From equation (1):

\[ a = 1 + 2b \quad (3) \]

Substitute equation (3) into equation (2):

\[ 2(1 + 2b) – 3b = 4 \]

\[ 2 + 4b – 3b = 4 \]

\[ b = 2 \]

Step 4: Find the Value of a

Substitute \(b = 2\) into equation (3):

\[ a = 1 + 2(2) = 5 \]

Conclusion

The required values are:

\[ a = 5,\quad b = 2 \]

\[ \therefore \quad \text{The values of } a \text{ and } b \text{ are } 5 \text{ and } 2. \]