What should be added to x^4+2x^3−2x^2+x−1 to make it exactly divisible by x^2+2x−3?

badlife18va

badlife18va

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2022-08-16

What should be added to x 4 + 2 x 3 2 x 2 + x 1 to make it exactly divisible by x 2 + 2 x 3 ?

Answer & Explanation

Bridget Vang

Bridget Vang

Beginner2022-08-17Added 11 answers

If you don't want to use polynomial long division, you can factor x 2 + 2 x 3 by seeing it has roots at 1 and -3, Then, you need to add a degree 1 polynomial to your polynomial such that the sum is divisible by x 1 and x + 3. Those are relatively prime, so the sum will then be divisible by the product, which is x 2 + 2 x 3. So,
x 2 + 2 x 3 x 4 + 2 x 3 2 x 2 + x 1 + a x + b   if ( x 1 ) ( x + 3 ) x 4 + 2 x 3 2 x 2 + x 1 + a x + b P ( 1 ) = P ( 3 ) = 0   where P ( x ) = x 4 + 2 x 3 2 x 2 + x 1 + a x + b P ( 1 ) = a + b + 1 = 0 , P ( 3 ) = 3 a + b + 5 = 0 a + b = 1 3 a + b = 5 a = 1 b = 2

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