Divide the following and express in the form [tex]P ( x )=[/tex] divisor [tex]x Q ( x )+ R[/tex], where R is a function of x.
a. [tex](x^3-x^2+3 x+2) \div(x^2-1)[/tex]
b. [tex](2 x^3+x^2-5 x-1) \div(x^2+3)[/tex]
c. [tex](5 x^4-x^2+2) \div(x^3-2)[/tex]

We performed polynomial division for three cases, providing quotients and remainders for each. The results are: a. P ( x ) = ( x 2 − 1 ) ( x − 1 ) + ( 4 x + 1 ) , b. P ( x ) = ( x 2 + 3 ) ( 2 x + 1 ) + ( − 11 x − 4 ) , c. P ( x ) = ( x 3 − 2 ) ( 5 x ) + ( − x 2 + 10 x + 2 ) .
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Answered by Anonymous | 2025-07-04

Perform polynomial division for (x^3 - x^2 + 3x + 2) / (x^2 - 1) and express the result as x 3 − x 2 + 3 x + 2 = ( x 2 − 1 ) ( x − 1 ) + ( 4 x + 1 ) .
Perform polynomial division for (2x^3 + x^2 - 5x - 1) / (x^2 + 3) and express the result as 2 x 3 + x 2 − 5 x − 1 = ( x 2 + 3 ) ( 2 x + 1 ) + ( − 11 x − 4 ) .
Perform polynomial division for (5x^4 - x^2 + 2) / (x^3 - 2) and express the result as 5 x 4 − x 2 + 2 = ( x 3 − 2 ) ( 5 x ) + ( − x 2 + 10 x + 2 ) .
The final results are: a. x 3 − x 2 + 3 x + 2 = ( x 2 − 1 ) ( x − 1 ) + ( 4 x + 1 ) , b. 2 x 3 + x 2 − 5 x − 1 = ( x 2 + 3 ) ( 2 x + 1 ) + ( − 11 x − 4 ) , c. 5 x 4 − x 2 + 2 = ( x 3 − 2 ) ( 5 x ) + ( − x 2 + 10 x + 2 ) .

Explanation

Problem Analysis We are given three polynomial division problems and asked to express each in the form P ( x ) = divisor × Q ( x ) + R ( x ) , where P ( x ) is the dividend, Q ( x ) is the quotient, and R ( x ) is the remainder. We will perform polynomial long division for each case.

Polynomial Division - Case a a. Divide ( x 3 − x 2 + 3 x + 2 ) by ( x 2 − 1 ) .
Using polynomial division, we find the quotient and remainder: x 3 − x 2 + 3 x + 2 = ( x 2 − 1 ) ( x − 1 ) + ( 4 x + 1 ) So, Q ( x ) = x − 1 and R ( x ) = 4 x + 1 .

Polynomial Division - Case b b. Divide ( 2 x 3 + x 2 − 5 x − 1 ) by ( x 2 + 3 ) .
Using polynomial division, we find the quotient and remainder: 2 x 3 + x 2 − 5 x − 1 = ( x 2 + 3 ) ( 2 x + 1 ) + ( − 11 x − 4 ) So, Q ( x ) = 2 x + 1 and R ( x ) = − 11 x − 4 .

Polynomial Division - Case c c. Divide ( 5 x 4 − x 2 + 2 ) by ( x 3 − 2 ) .
Using polynomial division, we find the quotient and remainder: 5 x 4 − x 2 + 2 = ( x 3 − 2 ) ( 5 x ) + ( − x 2 + 10 x + 2 ) So, Q ( x ) = 5 x and R ( x ) = − x 2 + 10 x + 2 .

Final Answer Therefore, the results of the polynomial divisions are: a. x 3 − x 2 + 3 x + 2 = ( x 2 − 1 ) ( x − 1 ) + ( 4 x + 1 ) b. 2 x 3 + x 2 − 5 x − 1 = ( x 2 + 3 ) ( 2 x + 1 ) + ( − 11 x − 4 ) c. 5 x 4 − x 2 + 2 = ( x 3 − 2 ) ( 5 x ) + ( − x 2 + 10 x + 2 )


Examples
Polynomial division is a fundamental concept in algebra with applications in various fields. For instance, in engineering, it can be used to simplify complex transfer functions in control systems. Imagine you have a system described by a rational function, and you want to analyze its behavior. By performing polynomial division, you can break down the function into simpler parts, making it easier to understand and design controllers for the system. This technique is also useful in signal processing for filter design and analysis.

Answered by GinnyAnswer | 2025-07-04