Perform the division.
$(60 x^{18}-54 x^{17}+30 x^{16}-18 x^{15}) \div(6 x^{19})$
Answer (2)
Divide each term of the polynomial by the monomial: 6 x 19 60 x 18 − 6 x 19 54 x 17 + 6 x 19 30 x 16 − 6 x 19 18 x 15 .
Simplify each term by dividing the coefficients and subtracting the exponents of x : 10 x − 1 − 9 x − 2 + 5 x − 3 − 3 x − 4 .
Rewrite the expression using positive exponents: x 10 − x 2 9 + x 3 5 − x 4 3 .
The final result is: x 10 − x 2 9 + x 3 5 − x 4 3 .
Explanation
Understanding the Problem We are asked to perform the division of the polynomial 60 x 18 − 54 x 17 + 30 x 16 − 18 x 15 by the monomial 6 x 19 . This involves dividing each term of the polynomial by the monomial.
Dividing Each Term We will divide each term of the polynomial by 6 x 19 : 6 x 19 60 x 18 − 54 x 17 + 30 x 16 − 18 x 15 = 6 x 19 60 x 18 − 6 x 19 54 x 17 + 6 x 19 30 x 16 − 6 x 19 18 x 15
Simplifying the Expression Now, we simplify each term by dividing the coefficients and subtracting the exponents of x :
6 60 x 18 − 19 − 6 54 x 17 − 19 + 6 30 x 16 − 19 − 6 18 x 15 − 19 = 10 x − 1 − 9 x − 2 + 5 x − 3 − 3 x − 4
Rewriting with Positive Exponents Finally, we rewrite the expression using positive exponents: 10 x − 1 − 9 x − 2 + 5 x − 3 − 3 x − 4 = x 10 − x 2 9 + x 3 5 − x 4 3
Final Result Therefore, the result of the division is: x 10 − x 2 9 + x 3 5 − x 4 3
Examples
Understanding polynomial division is crucial in various fields, such as engineering and computer science. For instance, when designing filters for signal processing, engineers use polynomial division to simplify transfer functions, making the filter easier to analyze and implement. Similarly, in computer graphics, polynomial division can be used to optimize rendering algorithms, improving the efficiency of displaying complex scenes. By mastering polynomial division, students gain a valuable tool applicable to real-world problems in technology and science.
To divide the polynomial 60 x 18 − 54 x 17 + 30 x 16 − 18 x 15 by 6 x 19 , we divide each term separately and simplify. The final result is x 10 − x 2 9 + x 3 5 − x 4 3 . Understanding this process is important for solving algebraic expressions more efficiently.
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