What value of $c$ makes the statement true?
$-2 x^3\left(c x^3+x^2\right)=10 x^6-2 x^5$
$c =$
Answer (1)
Expand the left side of the equation: − 2 x 3 ( c x 3 + x 2 ) = − 2 c x 6 − 2 x 5 .
Rewrite the equation: − 2 c x 6 − 2 x 5 = 10 x 6 − 2 x 5 .
Compare the coefficients of x 6 : − 2 c = 10 .
Solve for c : c = − 5 .
Explanation
Understanding the Problem We are given the equation − 2 x 3 ( c x 3 + x 2 ) = 10 x 6 − 2 x 5 and we need to find the value of c that makes this statement true.
Expanding the Left Side First, let's expand the left side of the equation by distributing the − 2 x 3 term: − 2 x 3 ( c x 3 + x 2 ) = − 2 x 3 ( c x 3 ) − 2 x 3 ( x 2 ) = − 2 c x 6 − 2 x 5
Rewriting the Equation Now we can rewrite the original equation with the expanded left side: − 2 c x 6 − 2 x 5 = 10 x 6 − 2 x 5
Comparing Coefficients To find the value of c , we need to compare the coefficients of the x 6 terms on both sides of the equation. The coefficient of x 6 on the left side is − 2 c , and the coefficient of x 6 on the right side is 10 . Therefore, we have: − 2 c = 10
Solving for c Now, we can solve for c by dividing both sides of the equation by − 2 :
c = − 2 10 = − 5
Final Answer Thus, the value of c that makes the statement true is − 5 .
Examples
Understanding polynomial equations and solving for unknown coefficients is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of an object, you might encounter an equation where you need to determine a specific parameter (like a damping coefficient) to match observed data. By expanding and comparing coefficients, you can find the value of the parameter that makes the equation accurately model the physical system. This technique is also used in control systems to design controllers that achieve desired system behavior.
