Condition for Divisibility of Polynomials

Video Explanation

Question

If the polynomial

\[ f(x) = ax^3 + bx – c \]

is divisible by

\[ g(x) = x^2 + bx + c, \]

find the value of \(ac\).

Solution

Step 1: Use the Division Algorithm

Since \(f(x)\) is divisible by \(g(x)\),

\[ f(x) = g(x)\,q(x) \]

where \(q(x)\) is a linear polynomial.

Let

\[ q(x) = kx + m \]

Step 2: Multiply the Polynomials

\[ (x^2 + bx + c)(kx + m) \]

\[ = kx^3 + (kb + m)x^2 + (kc + mb)x + mc \]

Step 3: Compare with \(f(x)\)

\[ ax^3 + 0x^2 + bx – c \]

Comparing coefficients:

\[ a = k \quad (1) \]

\[ kb + m = 0 \quad (2) \]

\[ kc + mb = b \quad (3) \]

\[ mc = -c \quad (4) \]

Step 4: Solve the Equations

From (4):

\[ m = -1 \]

Substitute in (2):

\[ kb – 1 = 0 \Rightarrow kb = 1 \]

From (1), \(k = a\), so:

\[ ab = 1 \quad (5) \]

Now use (3):

\[ kc + mb = b \]

\[ ac – b = b \]

\[ ac = 2b \]

From (5), \(b = \frac{1}{a}\).

So,

\[ ac = 2 \times \frac{1}{a} \times a = 2 \]

Conclusion

The required value of \(ac\) is:

\[ \boxed{2} \]