Divide and, if possible, simplify. (5 x+20)/(x^6) ÷(x+4)/(x^2)

Name: Problem 59, Chapter 6: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-07-13T03:38:43-08:00
Duration: 2 min 2 s
Channel: Ankit Gupta
Description: Divide and, if possible, simplify. (5 x+20)/(x^6) ÷(x+4)/(x^2)

step 1 5 times x plus 20 upon x to the power 6 multiply by reciprocal of divisor x square upon x plus 4

Divide and, if possible, simplify. (5 x+20)/(x^6)...

Key Concepts

Canceling Common Factors

After converting the division into multiplication and factoring the relevant polynomials, canceling common factors that appear in both the numerator and the denominator is the next essential step. This process reduces the expression to its simplest form by eliminating redundant parts of the fraction, subject to the condition that the cancelled factors do not equal zero, preserving the domain of the expression.

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. In the context of rational expressions, factoring is used to identify and extract common factors from the numerator or the denominator, making it easier to simplify the expression by canceling identical terms. This technique is especially useful when the expression includes binomials or common monomial factors.

Dividing Rational Expressions

Dividing rational expressions involves rewriting the division as a multiplication by the reciprocal of the divisor. This means that instead of dividing by a fraction, you multiply by its inverse. This step is fundamental because it converts the division problem into a multiplication problem, which is generally easier to handle and simplify.

Multiplying by the Reciprocal

When dividing by a fraction, take the reciprocal of the fraction you are dividing by. This means exchanging the numerator and the denominator of that fraction. This concept is central to handling any operation with fractions, as it allows us to transform division problems into multiplication ones, which are simpler to execute and combine with other algebraic operations.

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