Divide the following:
[tex]\begin{array}{l}
\frac{24 a^5 b^3 c^2-36 a^6 b^4 c^3+18 a^8 b^4 c^2}{6 a^2 b^3 c} \\
=\sqrt{x}-\sqrt{x}+\sqrt{x}
\end{array}[/tex]
Answer (1)
Divide each term of the polynomial by the monomial.
Simplify each resulting term using exponent rules.
Combine the simplified terms.
Simplify the right-hand side of the equation.
Square both sides to solve for x : x = ( 4 a 3 c − 6 a 4 b c 2 + 3 a 6 b c ) 2 .
Explanation
Understanding the Problem We are asked to divide a polynomial by a monomial and then solve for x . The given equation is: 6 a 2 b 3 c 24 a 5 b 3 c 2 − 36 a 6 b 4 c 3 + 18 a 8 b 4 c 2 = x − x + x
Dividing Each Term First, we simplify the left-hand side of the equation by dividing each term in the numerator by the denominator: 6 a 2 b 3 c 24 a 5 b 3 c 2 − 6 a 2 b 3 c 36 a 6 b 4 c 3 + 6 a 2 b 3 c 18 a 8 b 4 c 2
Simplifying the Terms Now, we simplify each term using the rules of exponents ( a n a m = a m − n ): 6 a 2 b 3 c 24 a 5 b 3 c 2 = 4 a 5 − 2 b 3 − 3 c 2 − 1 = 4 a 3 b 0 c 1 = 4 a 3 c 6 a 2 b 3 c 36 a 6 b 4 c 3 = 6 a 6 − 2 b 4 − 3 c 3 − 1 = 6 a 4 b 1 c 2 = 6 a 4 b c 2 6 a 2 b 3 c 18 a 8 b 4 c 2 = 3 a 8 − 2 b 4 − 3 c 2 − 1 = 3 a 6 b 1 c 1 = 3 a 6 b c So the left-hand side simplifies to: 4 a 3 c − 6 a 4 b c 2 + 3 a 6 b c
Simplifying the Right-Hand Side Next, we simplify the right-hand side of the equation: x − x + x = x So the equation becomes: 4 a 3 c − 6 a 4 b c 2 + 3 a 6 b c = x
Solving for x To solve for x , we square both sides of the equation: ( 4 a 3 c − 6 a 4 b c 2 + 3 a 6 b c ) 2 = ( x ) 2 x = ( 4 a 3 c − 6 a 4 b c 2 + 3 a 6 b c ) 2
Final Answer Therefore, the solution for x is: x = ( 4 a 3 c − 6 a 4 b c 2 + 3 a 6 b c ) 2
Examples
This type of polynomial division and simplification is fundamental in many areas of engineering and physics. For example, when analyzing complex circuits, engineers often simplify expressions to understand the behavior of the circuit. Similarly, in physics, simplifying equations helps in modeling physical phenomena more efficiently. Understanding how to manipulate and simplify these expressions is a crucial skill in these fields.
