Divide.
$\frac{12 u^7-8 u^5}{4 u^3}$
Simplify your answer as much as possible.
Answer (2)
∙ Divide each term in the numerator by the denominator: 4 u 3 12 u 7 − 4 u 3 8 u 5 .
∙ Simplify each term by dividing the coefficients and subtracting the exponents of u .
∙ 4 12 u 7 − 3 − 4 8 u 5 − 3 = 3 u 4 − 2 u 2 .
∙ The simplified expression is 3 u 4 − 2 u 2 .
Explanation
Understanding the Problem We are asked to divide the polynomial 12 u 7 − 8 u 5 by the monomial 4 u 3 . This involves dividing each term of the polynomial by the monomial and simplifying the result.
Dividing Each Term To divide the polynomial by the monomial, we divide each term of the polynomial by the monomial: 4 u 3 12 u 7 − 8 u 5 = 4 u 3 12 u 7 − 4 u 3 8 u 5
Simplifying Each Term Now, we simplify each term by dividing the coefficients and subtracting the exponents of u : 4 u 3 12 u 7 = 4 12 u 7 − 3 = 3 u 4 4 u 3 8 u 5 = 4 8 u 5 − 3 = 2 u 2
Final Result Substitute these simplified terms back into the expression: 4 u 3 12 u 7 − 4 u 3 8 u 5 = 3 u 4 − 2 u 2 So, the simplified expression is 3 u 4 − 2 u 2 .
Examples
Polynomial division is a fundamental concept in algebra and is used in various applications, such as simplifying complex expressions, solving equations, and modeling real-world phenomena. For instance, in physics, you might use polynomial division to simplify an equation that describes the motion of an object, making it easier to analyze and understand. Similarly, in engineering, polynomial division can be used to design filters or analyze the stability of a system. Understanding polynomial division helps in simplifying complex models and making accurate predictions.
To simplify 4 u 3 12 u 7 − 8 u 5 , divide each term in the numerator by the denominator, resulting in 3 u 4 − 2 u 2 . This is done by separating and simplifying each term's coefficients and exponents. The final answer is 3 u 4 − 2 u 2 .
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