Questions: Divide. (x^2+2x-35) ÷ (x-5)

Solution Steps

To divide the polynomial \(x^2 + 2x - 35\) by \(x - 5\), 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 formed until the degree of the remainder is less than the degree of the divisor.

We are tasked with dividing the polynomial \(x^2 + 2x - 35\) by \(x - 5\). We can express this as:

\[ \frac{x^2 + 2x - 35}{x - 5} \]

1. Divide the leading term of the dividend \(x^2\) by the leading term of the divisor \(x\): \[ \frac{x^2}{x} = x \]
2. Multiply the entire divisor \(x - 5\) by \(x\): \[ x(x - 5) = x^2 - 5x \]
3. Subtract this result from the original polynomial: \[ (x^2 + 2x - 35) - (x^2 - 5x) = 2x + 5x - 35 = 7x - 35 \]

1. Now, divide the leading term \(7x\) by the leading term of the divisor \(x\): \[ \frac{7x}{x} = 7 \]
2. Multiply the entire divisor \(x - 5\) by \(7\): \[ 7(x - 5) = 7x - 35 \]
3. Subtract this from the current polynomial: \[ (7x - 35) - (7x - 35) = 0 \]

Since the remainder is \(0\), we conclude that the division is exact. The quotient obtained from the division is:

\[ x + 7 \]

Final Answer