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How do you divide x22x15x+3 using polynomial long division?

Answer
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Hint: Here are the steps required for dividing by a polynomial containing more than one term using the long division method.
• Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term.
• Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol.
• Multiply the answer obtained in the previous step by the polynomial in front of the division symbol.
• Subtract and bring down the next term.
• Repeat steps 2, 3, and 4 until there are no more terms to bring down.
• Write the final answer. The term remaining after the last subtract step is the remainder and must be written as a fraction in the final answer.

Complete step by step answer:
In this question, we have to find the quotient and the remainder of the division ofx22x15, the dividend, by x+3, the divisor.
x+3)x22x15
The first step is that the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term. Here, the polynomial is already written in descending order and has not any missing term.
The second step is to divide the terms with the highest power inside the division symbol that isx2 by the term with the highest power outside the division symbol that is x. We will get the answer x.
x2x=x
The third step is to multiply the answer obtained in the previous step that is x by the polynomial in front of the division symbol iscx+3. We will get the answerx2+3x.
x(x+3)=x2+3x
Subtract and bring down the next term. We will get the answer5x15.
x+3)x22x15x
           ±x2±3x
           5x15
Again we will repeat steps 2, 3, and 4.
The second step is to divide the terms with the highest power inside the division symbol that is -5x by the term with the highest power outside the division symbol that is x. We will get the answer -5.
5xx=5
The third step is to multiply the answer obtained in the previous step that is -5 by the polynomial in front of the division symbol isx+3. We will get the answer5x15.
5(x+3)=5x15
Subtract and bring down the next term. We will get the answer 0.
x+3)x22x15x5
               ±x2±3x
                5x15
                5x15
                           0
Hence, the quotient is x5and the remainder is 0.

Note: We can verify our answer by the following formula.
Dividend=(Divisor×Quotient)+Remainder
Here, dividend isx22x15. Divisor isx+3. Quotient isx5. Remainder is 0.
Let us take the right-hand side.
RHS=(Divisor×Quotient)+Remainder
Let us put all the values.
RHS=(x5)×(x+3)+0 $$
Simplify the above term.
RHS=x2+3x5x15
That is equal to
 RHS=x22x15
Therefore,
RHS=LHS
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