Question

Divide. Divide \(d^{2}+15 d+45\) by \(d+5\)

Short answer

The quotient resulting from dividing \(d^{2}+15 d+45\) by \(d+5\) is \(d+10\), with a remainder of \(-5\).

Step-by-step solution

Recognize the Division

The problem asks to divide \(d^{2}+15 d+45\) by \(d+5\). This is a division problem involving polynomials. You can write the problem in long division format.

Divide the term with the highest power

Start by dividing the term with the highest degree in the numerator (\(d^2\)) by the term with the highest degree in the denominator (\(d\)). Thus, \(d^2/d = d\). Write \(d\) above the division bar.

Multiply and Subtract

Next, you multiply the divisor \(d+5\) by the result from step 2 (which is \(d\)) and subtract this from the original numerator. \(d \times (d+5) = d^2+5d\). When you subtract this from the original numerator, you get a new expression: \(10d+45\).

Repeat the process

Again, divide the new term with the highest degree by the term with the highest degree in the denominator. Thus, \(10d/d = 10\). Write this result next to the first one to get \(d+10\) as the result of the division so far.

Conclude the polynomial division

Repeat the process in step 3 with the result \(10\). You find out that \(10 \times (d+5) = 10d+50\), which after subtraction from the current numerator \(10d+45\), gives a remainder of -5. This means the division is complete.