Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Simplify the rational expression. $$\frac{x^{3}-10 x^{2}+31 x-30}{x-3}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(x^2 - 7x + 10\).

Step 1: Apply Polynomial Division

Perform the division of the numerator polynomial \(x^3 - 10x^2 + 31x -30\) by the denominator polynomial \(x - 3\). You may use long division or synthetic division to find the quotient.

Step 2: Do Long Division

Set up the long division as you would for numbers. The cubic term \(x^3\) is divided by \(x\) to get \(x^2\). Multiply \(x^2\) with the divisor \(x - 3\) to get \(x^3 - 3x^2\). Subtract this from the first two terms of the dividend \(x^3 - 10x^2\) to get \(-7x^2\). Bring down the next term \(31x\) from the dividend. Repeat the process by dividing \(-7x^2\) by \(x\) to get \(-7x\). Multiply \(-7x\) with \(x - 3\) to get \(-7x^2 + 21x\). Subtract to get the next remainder \(10x\). Finally, bring down the last term \(-30\) from the dividend, and repeat the process one more time to get the final quotient \(x^2 - 7x + 10\).

Step 3: Write Final Answer

The quotient obtained in step 2 is the simplified form of the given rational expression. So the simplified form of \(\frac{x^3 - 10x^2 + 31x - 30}{x - 3}\) is \(x^2 - 7x + 10\)

Key Concepts

Polynomial Long Division

Polynomial long division is similar to the long division process used with numbers, but it deals with dividing polynomials by other polynomials. Imagine you have to divide a three-term cubic polynomial by a linear polynomial. Start by dividing the highest degree term of the numerator by the highest degree term of the denominator. Write the result above the long division bar. Multiply this result by the denominator and write it under the numerator, then subtract. Repeat these steps until you reach a result that cannot be divided by the denominator.

This process helps to break complex algebraic fractions into simpler parts, which makes them easier to understand and work with. The result of long division is the quotient plus any remainder over the original divisor. However, when the degree of the remainder is less than the degree of the divisor, as in the exercise provided, the quotient itself is the simplified form of the original polynomial fraction.

Synthetic Division

Synthetic division is a shortcut method of polynomial division, especially when dividing by a linear factor. It's generally quicker and involves less writing than polynomial long division. Here's the basic idea: Write down the coefficients of the dividend polynomial. Then, for a divisor of the form \( x - c \), bring down the leading coefficient. Multiply this number by \( c \), put the result under the next coefficient, and add them. Repeat this pattern until you finish with all coefficients.

The resulting numbers give you the coefficients of the quotient polynomial. Synthetic division is especially handy for higher-degree polynomials as it saves a considerable amount of time and effort. However, it only works when the divisor is a linear polynomial with a leading coefficient of 1.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. An example is the cubic function in the given exercise. The degrees of these functions tell us the maximum number of solutions they can have. For instance, a cubic polynomial could have up to three roots. These functions are continuous and smooth, and they're defined for all real numbers, meaning they don't have breaks, holes, or sharp corners.

Oftentimes, we're interested in simplifying expressions involving polynomial functions or finding their roots, which are the values of the variable that make the function equal to zero. Understanding how to simplify polynomial expressions using division techniques is a critical algebraic technique for solving polynomial equations and for analysis in calculus such as finding limits.

Algebraic Techniques

Algebraic techniques encompass a variety of methods used to solve equations, simplify expressions, and manipulate algebraic terms. Techniques such as factoring, expanding expressions, and using properties of exponents are essential to any algebraic toolkit. In our exercise, simplifying the rational expression requires an understanding of polynomial division, whether using the long or synthetic method.

Mastering algebraic techniques allows you to solve a wide range of mathematical problems, from simple equations to more complex functions. It lays the foundation for higher mathematics, including calculus, and is instrumental in fields that use mathematical modeling, such as engineering, physics, and economics. For students, strengthening these skills means simplifying problems into manageable parts, making calculus and higher math much more approachable.