There is a polynomial which, when multiplied by x^2+2x+3, gives 2x^5+3x^4+8x^3+8x^2+18x+9. What is the polynomial?
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To find the polynomial that, when multiplied by x^2+2x+3, gives the given polynomial, we can use the process of polynomial long division. Let's go through the steps:
Step 1: Write the given polynomial in descending order.
2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9
Step 2: Divide the leading term of the given polynomial (2x^5) by the leading term of the divisor (x^2). The result is the leading term of the quotient.
2x^5 divided by x^2 is 2x^3.
Step 3: Multiply the entire divisor (x^2 + 2x + 3) by the quotient's leading term.
(2x^3) * (x^2 + 2x + 3) = 2x^5 + 4x^4 + 6x^3
Step 4: Subtract the result from Step 3 from the given polynomial. Combine like terms.
(2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9) - (2x^5 + 4x^4 + 6x^3) =
-1x^4 + 2x^3 + 8x^2 + 18x + 9
Step 5: Repeat steps 2-4 with the result from Step 4 as the new polynomial.
-1x^4 divided by x^2 is -1x^2.
(-1x^2) * (x^2 + 2x + 3) = -x^4 -2x^3 -3x^2
(-1x^4 + 2x^3 + 8x^2 + 18x + 9) - (-x^4 -2x^3 -3x^2)=
x^4 + 4x^2 + 18x + 9
Step 6: Repeat steps 2-4 with the result from Step 5 as the new polynomial.
x^4 divided by x^2 is x^2.
(x^2) * (x^2 + 2x + 3) = x^4 + 2x^3 + 3x^2
(x^4 + 4x^2 + 18x + 9) - (x^4 + 2x^3 + 3x^2) =
2x^3 + x^2 + 18x + 9
Step 7: Repeat steps 2-4 with the result from Step 6 as the new polynomial.
2x^3 divided by x^2 is 2x.
(2x) * (x^2 + 2x + 3) = 2x^3 + 4x^2 + 6x
(2x^3 + x^2 + 18x + 9) - (2x^3 + 4x^2 + 6x) =
-3x^2 + 12x + 9
Step 8: Repeat steps 2-4 with the result from Step 7 as the new polynomial.
-3x^2 divided by x^2 is -3.
(-3) * (x^2 + 2x + 3) = -3x^2 - 6x - 9
(-3x^2 + 12x + 9) - (-3x^2 - 6x - 9) =
18x + 18
Since the resulting polynomial (18x + 18) does not contain any x^2 term, we have completed the division process.
Step 9: Express the given polynomial as the product of the divisor (x^2 + 2x + 3) and the quotient obtained.
2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9 = (2x^3 - x^2 + 2x - 3) * (x^2 + 2x + 3)
Therefore, the polynomial is 2x^3 - x^2 + 2x - 3.