\frac{3 x^2-9}{x-3} \div \frac{x+3}{x}
Answer (1)
Rewrite the division as multiplication by the reciprocal: x โ 3 3 x 2 โ 9 โ โ
x + 3 x โ .
Factor the numerator: 3 x 2 โ 9 = 3 ( x 2 โ 3 ) .
Substitute the factored form into the expression: ( x โ 3 ) ( x + 3 ) 3 ( x 2 โ 3 ) x โ .
The simplified expression is ( x โ 3 ) ( x + 3 ) 3 x ( x 2 โ 3 ) โ .
Explanation
Understanding the Problem We are asked to simplify the expression x โ 3 3 x 2 โ 9 โ รท x x + 3 โ . This involves dividing one rational expression by another. To do this, we will rewrite the division as multiplication by the reciprocal. Then, we will factor the numerator of the first fraction and cancel any common factors. We must also consider the restrictions on x to avoid division by zero.
Rewrite as Multiplication First, rewrite the division as multiplication by the reciprocal: x โ 3 3 x 2 โ 9 โ รท x x + 3 โ = x โ 3 3 x 2 โ 9 โ โ
x + 3 x โ
Factor the Numerator Next, factor the numerator of the first fraction. We can factor out a 3 from both terms in the numerator: 3 x 2 โ 9 = 3 ( x 2 โ 3 ) Then, we can factor x 2 โ 3 as a difference of squares: x 2 โ 3 = ( x โ 3 โ ) ( x + 3 โ ) However, a more useful factorization in this case is to factor out 3 and then factor x 2 โ 3 as ( x โ 3 ) ( x + 3 ) :
3 x 2 โ 9 = 3 ( x 2 โ 3 ) = 3 ( x โ 3 โ ) ( x + 3 โ ) I made a mistake in the previous step. The correct factorization is: 3 x 2 โ 9 = 3 ( x 2 โ 3 ) However, we can also factor 3 x 2 โ 9 as 3 ( x โ 3 ) ( x + 3 ) if we made a mistake in the problem statement and the original expression was 3 x 2 โ 27 instead of 3 x 2 โ 9 . Let's proceed with the original problem.
Substitute Factored Form Substitute the factored form into the expression: x โ 3 3 ( x 2 โ 3 ) โ โ
x + 3 x โ = ( x โ 3 ) ( x + 3 ) 3 ( x 2 โ 3 ) x โ
Restrictions on x Now, let's consider the restrictions on x . We must have x โ 3 ๎ = 0 and x + 3 ๎ = 0 and x ๎ = 0 . Thus, x ๎ = 3 , x ๎ = โ 3 , and x ๎ = 0 .
Simplify the Expression The simplified expression is: ( x โ 3 ) ( x + 3 ) 3 ( x 2 โ 3 ) x โ If the original expression was x โ 3 3 x 2 โ 27 โ รท x x + 3 โ , then we would have: x โ 3 3 x 2 โ 27 โ รท x x + 3 โ = x โ 3 3 ( x 2 โ 9 ) โ โ
x + 3 x โ = x โ 3 3 ( x โ 3 ) ( x + 3 ) โ โ
x + 3 x โ = 3 x In this case, the simplified expression would be 3 x , provided x ๎ = 3 and x ๎ = โ 3 and x ๎ = 0 .
Final Answer Since the problem is x โ 3 3 x 2 โ 9 โ รท x x + 3 โ , the simplified expression is ( x โ 3 ) ( x + 3 ) 3 x ( x 2 โ 3 ) โ .
Examples
Rational expressions are used in various fields, such as physics, engineering, and economics. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves. Simplifying rational expressions makes it easier to analyze and understand these models.
