Problem 13 - Precalculus Mathematics for Calculus

Question

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 $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=2 x^{4}-x^{3}+9 x^{2}, \quad D(x)=x^{2}+4$$

Step-by-Step Solution

Answer

The quotient is \(2x^2 - x + 5\) and remainder is \(4x - 20\).

Step 1: Set Up the Division

We need to divide the polynomial \(P(x) = 2x^4 - x^3 + 9x^2\) by \(D(x) = x^2 + 4\). Since \(D(x)\) is not linear, we will use polynomial long division. Arrange the dividend and divisor in descending order of powers of \(x\) with placeholders for missing terms.

Step 2: Perform First Division Step

Compare the leading term of \(P(x) = 2x^4 - x^3 + 9x^2 + 0x + 0\) with that of \(D(x)\). Divide \(2x^4\) by \(x^2\) to get \(2x^2\). Multiply \(x^2 + 4\) by \(2x^2\) to obtain \(2x^4 + 8x^2\), and subtract this from \(P(x)\).New Polynomial: \(-x^3 + x^2\)

Step 3: Perform Second Division Step

Divide the new leading term \(-x^3\) by \(x^2\) to get \(-x\). Multiply \(x^2 + 4\) by \(-x\) to obtain \(-x^3 - 4x\), and subtract this from the remaining polynomial.New Polynomial: \(5x^2 + 4x\)

Step 4: Perform Third Division Step

Divide the leading term \(5x^2\) by \(x^2\) to get \(5\). Multiply \(x^2 + 4\) by \(5\) to obtain \(5x^2 + 20\), and subtract this from the remaining polynomial.New Polynomial: \(4x - 20\)

Step 5: Finalize the Remainder

The polynomial \(4x - 20\) is of degree lower than \(D(x) = x^2 + 4\), so it is our remainder. The quotient is therefore \(Q(x) = 2x^2 - x + 5\) and the remainder is \(R(x) = 4x - 20\).

Step 6: Express in Required Form

Using the formula \(\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}\), our division yields:\[\frac{2x^4 - x^3 + 9x^2}{x^2 + 4} = 2x^2 - x + 5 + \frac{4x - 20}{x^2 + 4}\]

Key Concepts

Polynomial Long Division

Polynomial Long Division is a method similar to the long division process used with numbers, but applied to polynomials. It involves dividing the terms of a polynomial dividend by the terms of a polynomial divisor.
To start, arrange both polynomials in decreasing order of their exponents, and make sure to include placeholders for any missing degrees.

• First, you divide the highest degree term of the dividend by the highest degree term of the divisor.
• Next, multiply the entire divisor by this result and subtract it from the dividend.
• Repeat these steps on the resulting polynomial, updating after each subtraction.

Continue this process until the remaining polynomial's degree is less than that of the divisor. The result consists of a quotient and a remainder, giving you a deeper understanding of polynomial expressions.

Synthetic Division

Synthetic Division is a simplified form of polynomial division, but it only works when the divisor is a linear polynomial. This technique reduces the appearances and complexity of coefficients, making it a faster alternative to long division. Here's how it works:

• Take the coefficients of the dividend polynomial as initial numbers in your synthetic division setup.
• Use the zero of the linear factor (divisor) and systematically apply it to transform these coefficients.
• Proceed with arithmetic operations, mainly multiplication and addition, repeatedly.

The results consist of new coefficients for your quotient. If you've covered all terms, and the remainder is non-zero, it will appear at the last step.

Quotient and Remainder

In polynomial division, the quotient and remainder concept is essential to express the full result of the division.When you divide a polynomial by another, you end up with a quotient and potentially a remainder. Mathematically, this is represented as:\[ \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} \]Where:

• Q(x) is the quotient polynomial, representing the whole part of the division.
• R(x) is the remainder polynomial, often with a degree lower than the divisor.

This format allows for easy understanding of how polynomials break down in arithmetic, similar to dividing numbers.

Polynomial Expressions

Polynomial Expressions are algebraic expressions made up of terms, which consist of variables raised to powers and coefficients.A typical polynomial is composed of terms like:- Coefficients, which could be real numbers.- Variables raised to powers.For example, in the polynomial \(2x^4 - x^3 + 9x^2\), each component represents a part of the larger expression.

• The term \(2x^4\) involves the coefficient 2 and the variable \(x\) raised to the power of 4.
• The properties such as degree, highest power, and number of terms help classify polynomials and determine the operations you can perform.

Understanding polynomials is crucial for operations like addition, subtraction, multiplication, and division, including the application of long and synthetic division processes.