Check whether g(t) = t² − 3 is a factor of f(t) = 2t⁴ + 3t³ − 2t² − 9t − 12 using division algorithm

Video Explanation

Watch the video explanation below:

Solution

Given:

g(t) = t² − 3
f(t) = 2t⁴ + 3t³ − 2t² − 9t − 12

Step 1: Apply the Division Algorithm

According to 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)

2t⁴ + 3t³ − 2t² − 9t − 12 ÷ (t² − 3)

First term:

2t⁴ ÷ t² = 2t²

Multiply:

2t²(t² − 3) = 2t⁴ − 6t²

Subtract:

(2t⁴ + 3t³ − 2t²) − (2t⁴ − 6t²)

= 3t³ + 4t²

Bring down −9t − 12:

3t³ + 4t² − 9t − 12

Next term:

3t³ ÷ t² = 3t

Multiply:

3t(t² − 3) = 3t³ − 9t

Subtract:

(3t³ + 4t² − 9t − 12) − (3t³ − 9t)

= 4t² − 12

Next term:

4t² ÷ t² = 4

Multiply:

4(t² − 3) = 4t² − 12

Subtract:

(4t² − 12) − (4t² − 12) = 0

Step 3: Identify Quotient and Remainder

Quotient, q(t) = 2t² + 3t + 4

Remainder, r(t) = 0

Final Answer

Since the remainder is zero, g(t) = t² − 3 is a factor of f(t).

Conclusion

Thus, by applying the division algorithm, we conclude that g(t) = t² − 3 is a factor of the polynomial f(t) = 2t⁴ + 3t³ − 2t² − 9t − 12.