Find the Values of a and b Using Factor Theorem

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question: If (x + 1) is a factor of 2x³ + ax² + 2bx + 1, find the values of a and b, given that 2a − 3b = 4.

Step 1: Use Factor Theorem

If (x + 1) is a factor, then f(−1) = 0

Let f(x) = 2x³ + ax² + 2bx + 1

f(−1) = 2(−1)³ + a(−1)² + 2b(−1) + 1

f(−1) = −2 + a − 2b + 1

f(−1) = a − 2b − 1

Since f(−1) = 0,

a − 2b − 1 = 0

⇒ a − 2b = 1  …… (1)

Step 2: Use the Given Condition

Given:

2a − 3b = 4  …… (2)

Step 3: Solve the System of Equations

Multiply equation (1) by 2:

2a − 4b = 2  …… (3)

Subtract equation (2) from equation (3):

(2a − 4b) − (2a − 3b) = 2 − 4

−b = −2

⇒ b = 2

Step 4: Find the Value of a

Substitute b = 2 in equation (1):

a − 2(2) = 1

a − 4 = 1

⇒ a = 5

Final Answer

∴ The required values are:

a = 5 and b = 2

Conclusion

Thus, by applying the Factor Theorem and solving the resulting system of equations, we find that the values of a and b are 5 and 2 respectively.