Questions: Divide. [ left(6 x^2+42 x+41right) div(x+6) ] Your answer should give the quotient and the remainder. Quotient: Remainder:

Solution Steps

We start with the polynomial \(6x^2 + 42x + 41\) as the dividend and \(x + 6\) as the divisor. We will perform polynomial long division to find the quotient and remainder.

We divide the leading term of the dividend \(6x^2\) by the leading term of the divisor \(x\), which gives us \(6x\). This is the first term of our quotient.

Next, we multiply the entire divisor \(x + 6\) by \(6x\) to get \(6x^2 + 36x\). We then subtract this from the original dividend: \[ (6x^2 + 42x + 41) - (6x^2 + 36x) = 6x + 41 \]

Now, we take the new polynomial \(6x + 41\) and repeat the process. We divide the leading term \(6x\) by \(x\) to get \(6\). This is the next term of our quotient.

We multiply the divisor \(x + 6\) by \(6\) to get \(6x + 36\). We subtract this from \(6x + 41\): \[ (6x + 41) - (6x + 36) = 5 \]

After completing the division, we find that the quotient is \(6x + 6\) and the remainder is \(5\). Thus, we can express the result of the division as: \[ \text{Quotient: } 6x + 6, \quad \text{Remainder: } 5 \]

Final Answer