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'''Without actual division prove that x⁴ + 2x³ -2x² + 2x - 3 is exactly divisible''' '''by x² + 2x - 3''' '''Solution:''' Let f(x) = x⁴ + 2x³ -2x² + 2x - 3 g(x) = x² + 2x - 3 = x(x + 3) - 1(x + 3) = (x - 1) (x + 3) Now f(x) will be exactly divisible by g(x) if it is exactly divisible by (x - 1) as well as (x + 3) i.e. if f(1) = 0 and f ( -3) = 0 Now f(1) = 1⁴ + 2.1³ -2.1² + 2.1 - 3 = 1 + 2 - 2 + 2 - 3 = 0 => (x - 1) is a factor of f(x) f ( -3) = (-3)⁴ + 2.(-3)³ -2.(-3)² + 2.(-3) - 3 = 81 - 54 - 18 - 6 - 3 = 0 => (x + 3) is a factor of f(x). => (x - 1) (x + 3) divides f (x) exactly Therefore, x² + 2x - 3 is a factor of f(x)

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