Check whether g(x) = 2x² − x + 3 is a factor of f(x) = 6x⁵ − x⁴ + 4x³ − 5x² − x − 15

Given

First polynomial:

g(x) = 2x² − x + 3

Second polynomial:

f(x) = 6x⁵ − x⁴ + 4x³ − 5x² − x − 15

To Check

Whether g(x) is a factor of f(x).

Method

We use the Division Algorithm.

If the remainder is zero, then g(x) is a factor of f(x).

Division Algorithm

f(x) = g(x) × q(x) + r(x)

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

Step 1: Divide f(x) by g(x)

6x⁵ − x⁴ + 4x³ − 5x² − x − 15 ÷ (2x² − x + 3)

First Term

6x⁵ ÷ 2x² = 3x³

Multiply:

3x³(2x² − x + 3) = 6x⁵ − 3x⁴ + 9x³

Subtract:

(6x⁵ − x⁴ + 4x³) − (6x⁵ − 3x⁴ + 9x³)

= 2x⁴ − 5x³

Bring down −5x²

2x⁴ − 5x³ − 5x²

Second Term

2x⁴ ÷ 2x² = x²

Multiply:

x²(2x² − x + 3) = 2x⁴ − x³ + 3x²

Subtract:

(2x⁴ − 5x³ − 5x²) − (2x⁴ − x³ + 3x²)

= −4x³ − 8x²

Bring down −x

−4x³ − 8x² − x

Third Term

−4x³ ÷ 2x² = −2x

Multiply:

−2x(2x² − x + 3) = −4x³ + 2x² − 6x

Subtract:

(−4x³ − 8x² − x) − (−4x³ + 2x² − 6x)

= −10x² + 5x

Bring down −15

−10x² + 5x − 15

Fourth Term

−10x² ÷ 2x² = −5

Multiply:

−5(2x² − x + 3) = −10x² + 5x − 15

Subtract:

(−10x² + 5x − 15) − (−10x² + 5x − 15) = 0

Step 2: Write Quotient and Remainder

Quotient, q(x) = 3x³ + x² − 2x − 5

Remainder, r(x) = 0

Final Answer

Since the remainder is zero, therefore

g(x) = 2x² − x + 3 is a factor of f(x).

Conclusion

Hence, the first polynomial divides the second polynomial exactly.