Checking Factor Using Division Algorithm

Video Explanation

Question

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

\[ g(x) = x^3 – 3x + 1, \quad f(x) = x^5 – 4x^3 + x^2 + 3x + 1 \]

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

\[ x^5 – 4x^3 + x^2 + 3x + 1 \]

by

\[ x^3 – 3x + 1 \]

First term

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

Subtract:

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

Bring down \(+3x + 1\).

Next term

\[ \frac{-x^3}{x^3} = -1 \]

Subtract:

\[ (-x^3 + 3x + 1) – (-x^3 + 3x – 1) = 2 \]

Step 3: Quotient and Remainder

\[ q(x) = x^2 – 1 \]

\[ r(x) = 2 \]

Conclusion

Since the remainder is not zero,

\[ \boxed{x^3 – 3x + 1 \text{ is not a factor of } x^5 – 4x^3 + x^2 + 3x + 1.} \]

Hence, the first polynomial is not a factor of the second polynomial.