Question 2:

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

(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12
(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2
(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

Answer:

(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12

    t² – 3 = t² + 0.t – 3

Since the remainder is 0,

Hence, t² – 3 is a factor of 2t⁴ + 3t³ – 2t² – 9t – 12.

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

Since the remainder is 0,

Hence, x² + 3x + 1 is a factor of 3x⁴ + 5x³ – 7x² + 2x + 2.

(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

Since the remainder ≠ 0,

Hence, x³ – 3x + 1 is not a factor of x⁵ – 4x³ + x² + 3x + 1.

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