Questions: Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x) / D(x) in the form P(x) / D(x) = Q(x) + R(x) / D(x). P(x) = 2 x^2 - 5 x - 6, D(x) = x - 2 P(x) / D(x) = □

Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express the quotient P(x) / D(x) in the form P(x) / D(x) = Q(x) + R(x) / D(x).

P(x) = 2 x^2 - 5 x - 6, D(x) = x - 2

P(x) / D(x) = □

Transcript text: Two polynomials $P$ and $D$ are given. Use either synthetic or long division to divide $P(x)$ by $D(x)$, and express the quotient $P(x) / D(x)$ in the form $\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$. \[ \begin{array}{l} P(x)=2 x^{2}-5 x-6, \quad D(x)=x-2 \\ \frac{P(x)}{D(x)}=\square \end{array} \]

Solution

Solution Steps

To divide the polynomial $ P(x) = 2x^2 - 5x - 6 $ by $ D(x) = x - 2 $, we can use synthetic division since the divisor is a linear polynomial of the form $ x - c $. The process involves using the root of the divisor, $ c = 2 $, to perform the division. The result will be a quotient polynomial $ Q(x) $ and a remainder $ R(x) $, allowing us to express the division in the form $ \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} $.

Step 1: Perform Synthetic Division

We start with the polynomial $ P(x) = 2x^2 - 5x - 6 $ and the divisor $ D(x) = x - 2 $. The root of the divisor is $ c = 2 $. Using synthetic division, we find the coefficients of the quotient and the remainder.

Step 2: Calculate Quotient and Remainder

After performing synthetic division, we obtain: