Question

Divide. Divide \(x^{2}-8 x+15\) by \(x-3\)

Short answer

The result of dividing the polynomial \(x^{2}-8 x+15\) by the binomial \(x-3\) is \(x - 5\).

Step-by-step solution

Divide First Term

The first step is to divide the first term in the numerator polynomial \(x^2\) by the first term in the denominator binomial \(x\). So, \(\frac{x^2}{x} = x\).

Multiply and Subtract

Now, the obtained result from Step 1 is multiplied with the divisor \(x-3\) to give \(x(x-3)\), which simplifies to \(x^2 - 3x\). Then, subtract this from the original polynomial, \((x^2 - 8x + 15) - (x^2 - 3x)\), to obtain the remainder after the first division, which is \(-8x + 3x + 15\) simplifying to \(-5x + 15\).

Divide again

Repeat Step 1, now with the first term of the new polynomial \(-5x\) by the first term of the divisor \(x\). \(\frac{-5x}{x} = -5\). Do the multiplication and subtraction also, this time getting \((-5x + 15) - (-5(x - 3)) = (-5x + 15) - (-5x + 15)\). So the remainder is 0. If there was a non-zero remainder, the process would be repeated until the degree of the remainder was less than the degree of the divisor.

Write the Result

After each step, combine the quotient from each division. In this case, it's \(x + (-5) = x - 5\), which is the final result since the remainder after the last division is 0. Thus, the polynomial \(x^{2}-8 x+15\) divided by the binomial \(x-3\) equals \(x-5\).