To perform polynomial division, we start by dividing the highest degree term of the dividend (x^3) by the highest degree term of the divisor (x). This gives us x^2 as the first term of the quotient. We then multiply the divisor (x+5) by this term, giving us (x^2)(x+5) = x^3+5x^2. We subtract this from the dividend:
x^3+5x^2+4x+18 - (x^3+5x^2) = 4x+18.
We then bring down the next term of the dividend, which is 4x, and repeat the process. Dividing 4x by x gives us 4 as the next term of the quotient. Multiplying the divisor (x+5) by this term gives us 4(x+5) = 4x+20. Subtracting this from the remaining part of the dividend:
4x+18 - (4x+20) = 4x+18-4x-20 = -2.
Finally, we bring down the last term of the dividend, which is 18. Dividing 18 by x gives us 18/x as the last term of the quotient. Multiplying the divisor (x+5) by this term gives us (18/x)(x+5) = 18+90/x. Subtracting this from the remaining part of the dividend:
-2 - (18+90/x) = -2-18-90/x = -20-90/x.
Therefore, the quotient (Q) is x^2 + 4 + 18/x, and the remainder (R) is -20-90/x, expressed as a fraction.
Consider the polynomial division (x^3+5x^2+4x+18)÷(x+5). Give the quotient with the remainder expressed as a fraction.
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