Questions: Find the quotient and remainder using long division for (x^2+3x-33)/(x-5)

Transcript text: Find the quotient and remainder using long division for $\frac{x^{2}+3 x-33}{x-5}$

Solution

Solution Steps

To find the quotient and remainder of the polynomial division $\frac{x^{2}+3x-33}{x-5}$, we can use polynomial long division. The process involves dividing the first term of the dividend by the first term of the divisor, multiplying the entire divisor by this result, subtracting from the dividend, and repeating the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Step 1: Define the Problem

We need to perform polynomial long division on the expression $\frac{x^{2}+3x-33}{x-5}$.

Step 2: Perform Polynomial Long Division

Divide the leading term of the dividend $x^2$ by the leading term of the divisor $x$ to get $x$.
Multiply the entire divisor $x - 5$ by $x$ to get $x^2 - 5x$.
Subtract this result from the original dividend: \[ (x^2 + 3x - 33) - (x^2 - 5x) = 8x - 33 \]

Step 3: Continue the Division

Now, divide the leading term $8x$ by the leading term of the divisor $x$ to get $8$.
Multiply the entire divisor $x - 5$ by $8$ to get $8x - 40$.
Subtract this from the current polynomial: \[ (8x - 33) - (8x - 40) = 7 \]

Step 4: Identify the Quotient and Remainder

The quotient from the division is $x + 8$ and the remainder is $7$.

Final Answer

The quotient is $x + 8$ and the remainder is $7$. Thus, we can express the final answer as: \[ \boxed{\text{Quotient: } x + 8, \text{ Remainder: } 7} \]

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