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Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3 as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Thenp(x)=___andp(x)q(x)=____

Short Answer

Expert verified

The required values are

p(x)=q(x)(x2-x+3)+(3x+1)andp(x)q(x)=(x2-x+3)+(3x+1)q(x)

Step by step solution

01

Step 1. Given Information

The given data is that we usepolynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder.

02

Step 2. Explanation

Suppose p(x)q(x)is an improper rational function where p(x) and q(x) are polynomial functions with deg(q(x))=m<n. Then,

p(x)q(x)=s(x)+r(x)q(x)

For some polynomial s(x) of degree n-mand some proper rational function r(x)q(x)with degr(x)<m

Thus, we get,

p(x)=q(x)(x2-x+3)+(3x+1)andp(x)q(x)=(x2-x+3)+(3x+1)q(x)

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