Problem 8 - 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 \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=2 x^{5}+4 x^{4}-4 x^{3}-x-3, \quad D(x)=x^{2}-2$$

Step-by-Step Solution

Answer

The quotient is \(2x^3 + 4x^2 + 2x + 8\) and the remainder is \(3x + 13\).

Step 1: Set Up the Division

We will use long division to divide the polynomial \(P(x) = 2x^5 + 4x^4 - 4x^3 - x - 3\) by \(D(x) = x^2 - 2\). Start by writing \(P(x)\) inside the long division bar and \(D(x)\) outside.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(2x^5\) by the leading term of the divisor \(x^2\). This gives us the first term of the quotient: \(2x^3\).

Step 3: Multiply and Subtract

Multiply \(2x^3\) by \(D(x) = x^2 - 2\), resulting in \(2x^5 - 4x^3\). Subtract this from the original dividend to get the new dividend, \(4x^4 + 2x^3 - x - 3\).

Step 4: Repeat the Division Process

Divide the new leading term \(4x^4\) by \(x^2\), yielding \(4x^2\). Multiply \(4x^2\) by \(x^2 - 2\) to get \(4x^4 - 8x^2\). Subtract to get \(2x^3 + 8x^2 - x - 3\).

Step 5: Continue Until Remainder's Degree is Less Than Divisor

Divide \(2x^3\) by \(x^2\), resulting in \(2x\). Multiply and subtract: \(2x imes (x^2 - 2) = 2x^3 - 4x\), leaving the remainder \(8x^2 + 3x - 3\).

Step 6: Final Division and Determine Remainder

Divide the leading term \(8x^2\) by \(x^2\), giving \(8\). Multiply and subtract: \(8 imes (x^2 - 2) = 8x^2 - 16\). The remainder is \(3x + 13\).

Step 7: Express in the Form of Division Equation

Finally, express \(P(x)\) in the form \(P(x) = D(x) \cdot Q(x) + R(x)\). Here, \(Q(x) = 2x^3 + 4x^2 + 2x + 8\) and \(R(x) = 3x + 13\).

Key Concepts

Long Division

Long division for polynomials is similar to the long division process you might remember from grade school arithmetic. The goal is to identify how many times the divisor (in this case, a polynomial) fits into the dividend, which is another polynomial.

Here’s a brief rundown of the steps:

• Start by organizing your polynomials in descending order of their degrees.
• Divide the leading term of the dividend by the leading term of the divisor. This will give you the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Repeat this process using the new polynomial (the remainder) as the dividend, until the degree of the remainder is less than the degree of the divisor.

Each time you repeat, you’re essentially peeling off another layer of the polynomial, until you can’t divide anymore due to the degree constraint. These steps will yield you both the quotient and the remainder.

Synthetic Division

Synthetic division is a simpler and faster method for dividing polynomials, but it's only applicable when the divisor is a linear polynomial of the form \(x - c\). This technique is known for its efficiency, especially when working with large polynomials or performing multiple divisions.

Here’s how synthetic division works:

• Write the coefficients of the dividend in a row. Include zeroes for any missing degrees in the polynomial.
• Use the zero of the divisor (i.e., \(c\) if the divisor is \(x - c\)) and perform synthetic division by a process similar to "bringing down" and "adding multiply back" until you're left with a new series of coefficients.
• The resulting coefficients represent the quotient, with the last coefficient being the remainder.

Synthetic division skips many intermediate steps that long division necessitates, making it a preferred choice for suitable problems.

Quotient and Remainder

When performing polynomial division, whether through long or synthetic methods, the result is expressed in terms of a quotient and a remainder. These terms are crucial

• The 'quotient' is what you get when you divide one polynomial by another completely, as much as possible.
• The 'remainder' is the leftover part of the dividend that the divisor can no longer divide entirely because its degree is less than that of the divisor.

In polynomial division, it’s important to note that the remainder will always be of a lesser degree than the divisor, ensuring the division process is complete. The division equation is then expressed as:
\[ P(x) = D(x) imes Q(x) + R(x) \]where \(P(x)\) is the original polynomial, \(D(x)\) is the divisor, \(Q(x)\) is the quotient, and \(R(x)\) is the remainder.

Polynomial Expressions

Polynomials are algebraic expressions that consist of terms in the form of \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer representing the degree. They are the building blocks of algebra, used in a variety of mathematical contexts.

Key characteristics of polynomial expressions include:

• Each term in a polynomial is a product of a number (coefficient) and a variable raised to a nonnegative integer power.
• The degree of the polynomial is determined by the highest power of the variable.
• Polynomials can be ordered in descending or ascending order based on their degrees.

Polynomial division, such as long and synthetic division, is just one of the ways these expressions are manipulated to solve complex algebraic problems. By understanding the structure and properties of polynomial expressions, students can effectively tackle algebraic equations and functions.