QuestionApril 19, 2025

Use long division to divide. Express any numbers as integers or simplified fractions. (x^5-2x^4+20x^3-15x+18)div (x^2-3)=square

Use long division to divide. Express any numbers as integers or simplified fractions. (x^5-2x^4+20x^3-15x+18)div (x^2-3)=square
Use long division to divide. Express any numbers as integers or simplified fractions. (x^5-2x^4+20x^3-15x+18)div (x^2-3)=square

Solution
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Answer

x^3 - 2x^2 + 23x - 6 + \frac{54x}{x^2 - 3} Explanation 1. Set up the division Arrange x^5 - 2x^4 + 20x^3 - 15x + 18 under the long division bar and x^2 - 3 outside. 2. Divide the leading terms Divide x^5 by x^2 to get x^3. Multiply x^3 by (x^2 - 3), resulting in x^5 - 3x^3. Subtract from the original polynomial. 3. Subtract and bring down next term Subtract x^5 - 3x^3 from x^5 - 2x^4 + 20x^3, resulting in -2x^4 + 23x^3. Bring down -15x. 4. Repeat division process Divide -2x^4 by x^2 to get -2x^2. Multiply -2x^2 by (x^2 - 3), resulting in -2x^4 + 6x^2. Subtract from current polynomial. 5. Subtract and bring down next term Subtract -2x^4 + 6x^2 from -2x^4 + 23x^3 - 15x, resulting in 23x^3 - 6x^2 - 15x. Bring down +18. 6. Continue division Divide 23x^3 by x^2 to get 23x. Multiply 23x by (x^2 - 3), resulting in 23x^3 - 69x. Subtract from current polynomial. 7. Subtract and finalize remainder Subtract 23x^3 - 69x from 23x^3 - 6x^2 - 15x + 18, resulting in -6x^2 + 54x + 18. 8. Final division step Divide -6x^2 by x^2 to get -6. Multiply -6 by (x^2 - 3), resulting in -6x^2 + 18. Subtract from current polynomial. 9. Conclude division Subtract -6x^2 + 18 from -6x^2 + 54x + 18, resulting in 54x. This is the remainder.

Explanation

1. Set up the division<br /> Arrange $x^5 - 2x^4 + 20x^3 - 15x + 18$ under the long division bar and $x^2 - 3$ outside.<br /><br />2. Divide the leading terms<br /> Divide $x^5$ by $x^2$ to get $x^3$. Multiply $x^3$ by $(x^2 - 3)$, resulting in $x^5 - 3x^3$. Subtract from the original polynomial.<br /><br />3. Subtract and bring down next term<br /> Subtract $x^5 - 3x^3$ from $x^5 - 2x^4 + 20x^3$, resulting in $-2x^4 + 23x^3$. Bring down $-15x$.<br /><br />4. Repeat division process<br /> Divide $-2x^4$ by $x^2$ to get $-2x^2$. Multiply $-2x^2$ by $(x^2 - 3)$, resulting in $-2x^4 + 6x^2$. Subtract from current polynomial.<br /><br />5. Subtract and bring down next term<br /> Subtract $-2x^4 + 6x^2$ from $-2x^4 + 23x^3 - 15x$, resulting in $23x^3 - 6x^2 - 15x$. Bring down $+18$.<br /><br />6. Continue division<br /> Divide $23x^3$ by $x^2$ to get $23x$. Multiply $23x$ by $(x^2 - 3)$, resulting in $23x^3 - 69x$. Subtract from current polynomial.<br /><br />7. Subtract and finalize remainder<br /> Subtract $23x^3 - 69x$ from $23x^3 - 6x^2 - 15x + 18$, resulting in $-6x^2 + 54x + 18$. <br /><br />8. Final division step<br /> Divide $-6x^2$ by $x^2$ to get $-6$. Multiply $-6$ by $(x^2 - 3)$, resulting in $-6x^2 + 18$. Subtract from current polynomial.<br /><br />9. Conclude division<br /> Subtract $-6x^2 + 18$ from $-6x^2 + 54x + 18$, resulting in $54x$. This is the remainder.
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