Questions: Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by P(x)=4 x^4-2 x^3+20 x^2, D(x)=2 x^2+9 P(x)/D(x)=

Transcript text: Two polynomials $P$ and $D$ are given. Use either synthetic or long division to divide $P(x)$ by \[ \begin{array}{l} P(x)=4 x^{4}-2 x^{3}+20 x^{2}, \quad D(x)=2 x^{2}+9 \\ \frac{P(x)}{D(x)}= \end{array} \]

Solution

Solution Steps

To divide the polynomial $ P(x) = 4x^4 - 2x^3 + 20x^2 $ by $ D(x) = 2x^2 + 9 $, we can use polynomial long division. The process involves dividing the leading term of the numerator by the leading term of the denominator, multiplying the entire divisor by this result, subtracting it from the original polynomial, and repeating the process with the remainder until the degree of the remainder is less than the degree of the divisor.

Step 1: Perform Polynomial Division

To divide the polynomial $ P(x) = 4x^4 - 2x^3 + 20x^2 $ by $ D(x) = 2x^2 + 9 $, we use polynomial long division. The quotient and remainder are obtained as follows:

Step 2: Identify Quotient and Remainder

From the division, we get: \[ \text{Quotient} = 2x^2 - x + 1 \] \[ \text{Remainder} = 9x - 9 \]

Step 3: Write the Final Expression

The division of $ P(x) $ by $ D(x) $ can be expressed as: \[ \frac{P(x)}{D(x)} = 2x^2 - x + 1 + \frac{9x - 9}{2x^2 + 9} \]