Trying to find the right answer to this..

x 2 + 2x - 3 = x 2 + 3x - x - 3 = x(x + 3) - 1 (x + 3) = (x + 3) (x - 1) Let p(x) = x 4 + 2x 3 - 2x 2 + 2x - 3 We see that p(-3) = (-3) 4 + 2(-3) 3 - 2(-3) 2 + 2(-3) - 3 = 81 - 54 - 18 - 6 - 3 = 0 Hence by converse of factor theorem, (x + 3) is a factor of p(x). Also, we see that p(1) = (1) 4 + 2(1) 3 - 2(1) 2 + 2(1) - 3 = 0 Hence by converse of factor theorem, (x - 1) is a factor of p(x). From above, we see that (x + 3) (x - 1), i.e., x 2 + 2x - 3 is a factor of p(x) ⇒ p(x) is exactly divisible by (x 2 + 2x - 3).

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Polynomials

Short Answer Type

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126. Without actual division prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.

x² + 2x - 3
= x² + 3x - x - 3
= x(x + 3) - 1 (x + 3)
= (x + 3) (x - 1)
Let p(x) = x⁴ + 2x³ - 2x² + 2x - 3
We see that
p(-3) = (-3)⁴ + 2(-3)³ - 2(-3)²+ 2(-3) - 3
= 81 - 54 - 18 - 6 - 3
= 0
Hence by converse of factor theorem, (x + 3) is a factor of p(x).
Also, we see that
p(1) = (1)⁴ + 2(1)³ - 2(1)² + 2(1) - 3
= 0
Hence by converse of factor theorem, (x - 1) is a factor of p(x).
From above, we see that
(x + 3) (x - 1), i.e., x² + 2x - 3 is a factor of p(x)
⇒ p(x) is exactly divisible by (x² + 2x - 3).