Checking Factor Using Division Algorithm

Video Explanation

Question

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

\[ g(t) = t^2 – 3, \quad f(t) = 2t^4 + 3t^3 – 2t^2 – 9t – 12 \]

Solution

Step 1: Apply the Division Algorithm

By the division algorithm,

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

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

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

Divide

\[ 2t^4 + 3t^3 – 2t^2 – 9t – 12 \]

by

\[ t^2 – 3 \]

First term

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

Subtract:

\[ (2t^4 + 3t^3 – 2t^2) – (2t^4 – 6t^2) = 3t^3 + 4t^2 \]

Bring down \(-9t\).

Next term

\[ \frac{3t^3}{t^2} = 3t \]

Subtract:

\[ (3t^3 + 4t^2 – 9t) – (3t^3 – 9t) = 4t^2 \]

Bring down \(-12\).

Next term

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

Subtract:

\[ (4t^2 – 12) – (4t^2 – 12) = 0 \]

Step 3: Identify Quotient and Remainder

\[ q(t) = 2t^2 + 3t + 4 \]

\[ r(t) = 0 \]

Conclusion

Since the remainder obtained is zero,

\[ \boxed{t^2 – 3 \text{ is a factor of } 2t^4 + 3t^3 – 2t^2 – 9t – 12.} \]

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