Chapter 1.4, Problem 4E

(a)

To determine

To find: The number of terms in the expression.

Answer to Problem 4E

The number of terms is 3 in the expression.

Explanation of Solution

A polynomial with variable x is an expression of the form,

$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$

Where,

• a is a real number.
• n is nonnegative integer.

The monomial $a_n{x}^n$ that combine together to make the polynomial are known as the terms of the polynomial.

The given expression is,

$\frac{1}{x}-\frac{2}{x+1}-\frac{x}{{\left(x+1\right)}^2}$

Thus, the number of terms is 3 in the expression.

(b)

To determine

To find: The least common denominator of the terms.

Answer to Problem 4E

The least common denominator of all the terms is $x{\left(x+1\right)}^2$.

Explanation of Solution

Given:

The given expression is,

$\frac{1}{x}-\frac{2}{x+1}-\frac{x}{{\left(x+1\right)}^2}$

Calculation:

Factor each denominator of given expression and take the product of distinct factors, than the highest power that appears in any of the factors is the least common denominator.

All the denominators of the expression are x, $\left(x+1\right)$ and ${\left(x+1\right)}^2$. The least common denominator is $x{\left(x+1\right)}^2$.

Thus, the least common denominator of all the terms is $x{\left(x+1\right)}^2$.

(c)

To determine

To find: The sum of given rational fractions.

Answer to Problem 4E

The sum of given rational fractions is $\frac{-2x^2+1}{x{\left(x+1\right)}^2}$.

Explanation of Solution

Given:

The expression is,

$\frac{1}{x}-\frac{2}{x+1}-\frac{x}{{\left(x+1\right)}^2}$

Calculation:

Use least common denominator to add the given rational fractions.

$\begin{array}{c}\frac{1}{x}-\frac{2}{x+1}-\frac{x}{{\left(x+1\right)}^2}=\frac{{\left(x+1\right)}^2}{x{\left(x+1\right)}^2}-\frac{2x\left(x+1\right)}{x{\left(x+1\right)}^2}-\frac{x^2}{x{\left(x+1\right)}^2}\\ =\frac{x^2+2x+1}{x{\left(x+1\right)}^2}-\frac{2x^2+2x}{x{\left(x+1\right)}^2}-\frac{x^2}{x{\left(x+1\right)}^2}\\ =\frac{x^2+2x+1-2x^2-2x-x^2}{x{\left(x+1\right)}^2}\left(\text{Combine}\text{terms}\text{in}\text{numerator}\right)\\ =\frac{-2x^2+1}{x{\left(x+1\right)}^2}\end{array}$

Thus, the sum of given rational fractions is $\frac{-2x^2+1}{x{\left(x+1\right)}^2}$.