$y=\frac{\left(x^2+x\right)\left(x^{-2}+1\right)}{x^2}$
Answer (2)
The expression y = x 2 ( x 2 + x ) ( x − 2 + 1 ) can be simplified by expanding the numerator and dividing by x 2 . The final simplified form is y = 1 + x 1 + x 2 1 + x 3 1 . This technique demonstrates the importance of understanding algebraic manipulation in mathematics.
;
Expand the numerator: ( x 2 + x ) ( x − 2 + 1 ) = 1 + x 2 + x − 1 + x .
Divide each term by x 2 : y = x 2 1 + x 2 x 2 + x 2 x − 1 + x 2 x .
Simplify the expression: y = x − 2 + 1 + x − 3 + x − 1 .
Rewrite with positive exponents and rearrange: y = 1 + x 1 + x 2 1 + x 3 1 .
The simplified equation is y = 1 + x 1 + x 2 1 + x 3 1 .
Explanation
Understanding the Problem We are given the equation y = x 2 ( x 2 + x ) ( x − 2 + 1 ) and we want to simplify it.
Expanding the Numerator First, let's expand the numerator. We have ( x 2 + x ) ( x − 2 + 1 ) = x 2 ( x − 2 ) + x 2 ( 1 ) + x ( x − 2 ) + x ( 1 ) = 1 + x 2 + x − 1 + x . So, the equation becomes y = x 2 1 + x 2 + x − 1 + x .
Dividing by x^2 Now, we divide each term in the numerator by x 2 :
y = x 2 1 + x 2 x 2 + x 2 x − 1 + x 2 x = x − 2 + 1 + x − 3 + x − 1 .
Rewriting with Positive Exponents Rewriting the expression with positive exponents, we get y = x 2 1 + 1 + x 3 1 + x 1 .
Rearranging the Terms Finally, we rearrange the terms to get y = 1 + x 1 + x 2 1 + x 3 1 .
Final Answer Therefore, the simplified equation is y = 1 + x 1 + x 2 1 + x 3 1 .
Examples
Understanding how to simplify rational expressions is crucial in many fields, such as physics and engineering, where complex equations often need to be manipulated to solve for unknown variables. For example, in circuit analysis, simplifying expressions involving impedances can help engineers design more efficient circuits. Similarly, in fluid dynamics, simplifying equations describing fluid flow can aid in predicting the behavior of liquids and gases under various conditions. By mastering these algebraic techniques, students can tackle real-world problems with greater confidence and precision.
