Checking Factor Using Division Algorithm

Video Explanation

Question

Check whether the first polynomial is a factor of the second polynomial:

\[ g(x) = 2x^2 – x + 3, \quad f(x) = 6x^5 – x^4 + 4x^3 – 5x^2 – x – 15 \]

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 \(g(x)\)

Divide

\[ 6x^5 – x^4 + 4x^3 – 5x^2 – x – 15 \]

by

\[ 2x^2 – x + 3 \]

First term

\[ \frac{6x^5}{2x^2} = 3x^3 \]

Subtract:

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

Bring down \(-5x^2\).

Next term

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

Subtract:

\[ (2x^4 – 5x^3 – 5x^2) – (2x^4 – x^3 + 3x^2) = -4x^3 – 8x^2 \]

Bring down \(-x\).

Next term

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

Subtract:

\[ (-4x^3 – 8x^2 – x) – (-4x^3 + 2x^2 – 6x) = -10x^2 + 5x \]

Bring down \(-15\).

Next term

\[ \frac{-10x^2}{2x^2} = -5 \]

Subtract:

\[ (-10x^2 + 5x – 15) – (-10x^2 + 5x – 15) = 0 \]

Step 3: Quotient and Remainder

\[ q(x) = 3x^3 + x^2 – 2x – 5 \]

\[ r(x) = 0 \]

Conclusion

Since the remainder obtained is zero,

\[ \boxed{2x^2 – x + 3 \text{ is a factor of } 6x^5 – x^4 + 4x^3 – 5x^2 – x – 15.} \]

Hence, the first polynomial is a factor of the second polynomial.