Use long division to divide. Specify the quotient and the remainder.
[tex]$\left(x^3-521\right) \div(x-8)$[/tex]
quotient $\square$
remainder $\square$
Answer (2)
Using polynomial long division to divide x 3 − 521 by x − 8 , we find the quotient is x 2 + 8 x + 64 and the remainder is − 9 .
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Perform polynomial long division of x 3 − 521 by x − 8 .
Divide x 3 by x to get the first term of the quotient, x 2 .
Continue the long division process to find the quotient x 2 + 8 x + 64 and the remainder − 9 .
State the quotient and remainder: x 2 + 8 x + 64 and − 9 .
Explanation
Understanding the Problem We want to divide the polynomial x 3 − 521 by the polynomial x − 8 using long division. Our goal is to find the quotient and the remainder of this division.
Setting up Long Division Let's set up the long division. We write x 3 − 521 as x 3 + 0 x 2 + 0 x − 521 to keep track of the place values.
First Step of Division First, divide x 3 by x to get x 2 . This is the first term of the quotient. Multiply x 2 by ( x − 8 ) to get x 3 − 8 x 2 . Subtract this from x 3 + 0 x 2 to get 8 x 2 . Bring down the next term, 0 x , to get 8 x 2 + 0 x .
Second Step of Division Next, divide 8 x 2 by x to get 8 x . This is the second term of the quotient. Multiply 8 x by ( x − 8 ) to get 8 x 2 − 64 x . Subtract this from 8 x 2 + 0 x to get 64 x . Bring down the next term, − 521 , to get 64 x − 521 .
Third Step of Division Now, divide 64 x by x to get 64 . This is the third term of the quotient. Multiply 64 by ( x − 8 ) to get 64 x − 512 . Subtract this from 64 x − 521 to get − 9 .
Final Result Therefore, the quotient is x 2 + 8 x + 64 and the remainder is − 9 .
Examples
Polynomial long division is a method for dividing one polynomial by another polynomial of a lower degree. It's similar to long division with numbers. For example, if you're designing a rectangular garden and know the area can be expressed as x 3 − 521 square feet, and you want one side to be x − 8 feet long, polynomial long division helps you find the expression for the length of the other side ( x 2 + 8 x + 64 feet) and determine if there's any leftover area (remainder of − 9 square feet, which might indicate an adjustment needed).
