Finding the Value of k Using Division Algorithm

Video Explanation

Question

For what value of \(k\) is the polynomial

\[ f(x) = 3x^4 – 9x^3 + x^2 + 15x + k \]

completely divisible by

\[ g(x) = 3x^2 – 5 \; ? \]

Solution

Step 1: Apply the Division Algorithm

By the division algorithm,

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

where the degree of \(r(x)\) is less than the degree of \(g(x)\).

Step 2: Divide \(f(x)\) by \(g(x)\)

First term

\[ \frac{3x^4}{3x^2} = x^2 \]

Subtract:

\[ (3x^4 – 9x^3 + x^2) – (3x^4 – 5x^2) = -9x^3 + 6x^2 \]

Bring down \(+15x\).

Next term

\[ \frac{-9x^3}{3x^2} = -3x \]

Subtract:

\[ (-9x^3 + 6x^2 + 15x) – (-9x^3 + 15x) = 6x^2 \]

Bring down \(+k\).

Next term

\[ \frac{6x^2}{3x^2} = 2 \]

Subtract:

\[ (6x^2 + k) – (6x^2 – 10) = k + 10 \]

Step 3: Use the Condition for Complete Divisibility

For complete divisibility,

\[ r(x) = 0 \]

So,

\[ k + 10 = 0 \Rightarrow k = -10 \]

Conclusion

The value of \(k\) for which the polynomial

\[ 3x^4 – 9x^3 + x^2 + 15x + k \]

is completely divisible by

\[ 3x^2 – 5 \]

is

\[ \boxed{k = -10} \]