Multiply and, if possible, simplify. (x^2-6 x+9)/(12-4 x) ·(x^6-9 x^4)/(x^3-3 x^2)

Name: Problem 52, Chapter 6: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-07-13T03:37:41-08:00
Duration: 2 min 25 s
Channel: Ankit Gupta
Description: Multiply and, if possible, simplify. (x^2-6 x+9)/(12-4 x) ·(x^6-9 x^4)/(x^3-3 x^2)

step 1 x minus 3 times x minus 3 upon minus 4 times x minus 3 multiply by x to the power 4 time x squar

Multiply and, if possible, simplify. (x^2-6 x+9)/(12-4 x)...

Key Concepts

Domain Restrictions

When simplifying rational expressions, it is essential to consider the domain restrictions. Values that make any original denominator equal to zero must be excluded from the solution set, even if they appear to cancel out in the process of simplification. This ensures that any simplified expression remains equivalent to the original expression in terms of its valid inputs.

Factoring Polynomials

Factoring polynomials is the process of rewriting a polynomial as a product of its factors. Recognizing patterns such as perfect square trinomials, differences of squares, or common factors is crucial. Factoring can simplify complex expressions and is a key step in the simplification of rational expressions.

Simplification by Cancelling Common Factors

Simplification of rational expressions involves canceling common factors in the numerator and the denominator. This step reduces the expression to its simplest form, as long as care is taken to avoid canceling factors that are equal to zero, which would affect the domain of the expression.

Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. They are similar to numerical fractions and can be manipulated using similar rules, including multiplication, division, addition, and subtraction, provided that any restrictions (such as values that make the denominator zero) are noted.

Multiplying Fractions

Multiplying rational expressions involves multiplying the numerators together and the denominators together. This process relies on the basic property of multiplication for fractions and is often followed by the simplification of the resulting expression.

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