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Chapter 11: Q. 12 (page 765)

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

2x(x2+4)x2+1.

Short Answer

Expert verified

The rational expression 2x(x2+4)x2+1is improper.

The sum of a polynomial and a proper rational expression isrole="math" localid="1647447232212" 2x+6xx2+1.

Step by step solution

01

Step 1. Given Information.

The rational expression is,

2x(x2+4)x2+1.

02

Step 2. Simplifying the rational expression.

The rational expression is,

2x(x2+4)x2+1

Simplifying the expression,

=2x3+8xx2+1.

03

Step 3. Improper or Proper Rational Expression.

2x3+8xx2+1

The degree of the numerator=3.

The degree of the denominator=2.

The degree of the numerator > The degree of the denominator.

It is an improper rational expression.

04

Step 4. Decomposition of Rational Expression.

The improper rational expression is,

2x3+8xx2+1.

Splitting the term and solving,

=2x(x2+4)x2+1=2x(x2+1+3)x2+1=2x(x2+1)x2+1+6xx2+1=2x+6xx2+1

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