Answer each question. By what expression should we multiply each side of (3x - 2)/(x + 4)(3x^2 + 1) = A/(x + 4) + (Bx + C)/(3x^2 + 1) so that there are no fractions in the equation?
Verified step by step guidance
1
Identify the denominators in the equation: , , and .
To eliminate the fractions, multiply every term in the equation by the least common denominator (LCD), which is .
Multiply the left side of the equation by to cancel out the denominator.
Multiply the right side terms: by and by to cancel their respective denominators.
After multiplying, simplify each term to remove the fractions, resulting in a polynomial equation.
Recommended similar problem, with video answer:
Verified Solution
This video solution was recommended by our tutors as helpful for the problem above
Video duration:
4m
Play a video:
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Common Denominator
A common denominator is a shared multiple of the denominators in a set of fractions. In this context, to eliminate fractions from the equation, we need to identify the least common denominator (LCD) of the fractions involved, which is the product of the unique factors of each denominator. By multiplying both sides of the equation by the LCD, we can simplify the equation to a polynomial form without fractions.
Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is crucial for solving equations involving them. In this problem, we are dealing with rational expressions that require careful handling of their components to combine or simplify them effectively.
Algebraic manipulation involves applying various algebraic techniques to rearrange and simplify expressions or equations. This includes operations such as factoring, distributing, and combining like terms. In the context of this question, algebraic manipulation is necessary to transform the equation into a form that is easier to solve, particularly after eliminating the fractions.