For what value of k is the polynomial f(x) = 3x⁴ − 9x³ + x² + 15x + k completely divisible by 3x² − 5

Video Explanation

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Given

f(x) = 3x⁴ − 9x³ + x² + 15x + k

Divisor = 3x² − 5

To Find

The value of k for which f(x) is completely divisible by 3x² − 5.

Concept Used

If a polynomial f(x) is completely divisible by another polynomial g(x), then the remainder obtained on division is zero.

According to the division algorithm:

f(x) = g(x) · q(x) + r(x)

Solution

Step 1: Divide f(x) by 3x² − 5

3x⁴ − 9x³ + x² + 15x + k ÷ (3x² − 5)

First term:

3x⁴ ÷ 3x² = x²

Multiply:

x²(3x² − 5) = 3x⁴ − 5x²

Subtract:

(3x⁴ − 9x³ + x²) − (3x⁴ − 5x²)

= −9x³ + 6x²

Bring down +15x:

−9x³ + 6x² + 15x

Next term:

−9x³ ÷ 3x² = −3x

Multiply:

−3x(3x² − 5) = −9x³ + 15x

Subtract:

(−9x³ + 6x² + 15x) − (−9x³ + 15x)

= 6x²

Bring down +k:

6x² + k

Next term:

6x² ÷ 3x² = 2

Multiply:

2(3x² − 5) = 6x² − 10

Subtract:

(6x² + k) − (6x² − 10)

= k + 10

Step 2: Use the Condition for Complete Divisibility

For f(x) to be completely divisible by 3x² − 5,

Remainder = 0

∴ k + 10 = 0

∴ k = −10

Final Answer

The value of k for which the polynomial is completely divisible by 3x² − 5 is:

k = −10

Conclusion

Hence, the polynomial f(x) = 3x⁴ − 9x³ + x² + 15x + k is completely divisible by 3x² − 5 when k = −10.