If the polynomial f(x) = ax³ + bx − c is divisible by g(x) = x² + bx + c, find the value of ac

Video Explanation

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Given

f(x) = ax³ + bx − c

g(x) = x² + bx + c

f(x) is divisible by g(x).

To Find

The value of ac.

Solution

Step 1: Use the Division Algorithm

Since f(x) is divisible by g(x), the remainder is zero.

Let the quotient be (ax + k).

Then:

f(x) = (x² + bx + c)(ax + k)

Step 2: Multiply

(ax + k)(x² + bx + c)

= ax³ + abx² + acx + kx² + kbx + kc

= ax³ + (ab + k)x² + (ac + kb)x + kc

Step 3: Compare with f(x)

Given:

f(x) = ax³ + 0x² + bx − c

Comparing coefficients:

ab + k = 0   …(1)

ac + kb = b   …(2)

kc = −c   …(3)

Step 4: Solve

From (3):

kc = −c

⇒ k = −1

Substitute k = −1 in (2):

ac − b = b

⇒ ac = 2b

Final Answer

ac = 2b

Conclusion

Hence, if the polynomial f(x) = ax³ + bx − c is divisible by g(x) = x² + bx + c, then the value of ac is 2b.