Problem 101 - Intermediate Algebra

Question

Perform each division. \(\frac{c^{4}-c^{2} d^{2}+10 c^{2}-6 d^{2}+23}{c^{2}+6}\)

Step-by-Step Solution

Answer

The quotient is \(c^2 - d^2 + 4\) with a remainder of \(-1\).

Step 1: Identify the Dividend and the Divisor

The expression to be divided is the dividend, which is \(c^{4}-c^{2} d^{2}+10 c^{2}-6 d^{2}+23\). The divisor is \(c^2 + 6\).

Step 2: Divide the Leading Terms

First, divide the leading term of the dividend \(c^4\) by the leading term of the divisor \(c^2\). This gives us \(c^2\). This will be our first term in the quotient.

Step 3: Multiply and Subtract

Multiply \(c^2\) by \(c^2 + 6\) to get \(c^4 + 6c^2\). Subtract this result from the original dividend, \(c^4 - c^2d^2 + 10c^2 - 6d^2 + 23\). This gives the new dividend: \(-c^2d^2 + 4c^2 - 6d^2 + 23\).

Step 4: Repeat the Division Process

Now, divide the new leading term \(-c^2d^2\) by the leading term of the divisor \(c^2\), which results in \(-d^2\). Add this to the quotient, resulting in \(c^2 - d^2\).

Step 5: Multiply and Subtract Again

Multiply \(-d^2\) by \(c^2 + 6\) to obtain \(-c^2d^2 - 6d^2\). Subtract this from the new dividend \(-c^2d^2 + 4c^2 - 6d^2 + 23\) to get the new dividend \(4c^2 + 23\).

Step 6: Final Division and Obtain Remainder

Divide \(4c^2\) by \(c^2\), yielding \(4\). Multiply \(4\) by \(c^2 + 6\) to obtain \(4c^2 + 24\). Subtract this from the current dividend \(4c^2 + 23\), leaving the remainder \(-1\).

Step 7: Write the Final Result

The final result of dividing \(c^{4}-c^{2} d^{2}+10 c^{2}-6 d^{2}+23\) by \(c^{2}+6\) is \(c^2 - d^2 + 4\), with a remainder of \(-1\). Thus, the expression can be written as \(c^2 - d^2 + 4 + \frac{-1}{c^2 + 6}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators such as addition, subtraction, multiplication, and division. These expressions form the basic building blocks of algebra. Understanding how to manipulate these expressions is crucial in solving complex mathematical problems.

• Variables: Symbols that stand for unknown quantities. In the given expression, \(c\) and \(d\) are variables.
• Constants: Fixed values. In expressions, these are usually numbers like 10 or 23.
• Terms: A single mathematical entity. It could be a constant, a variable, or both combined by multiplication or division, such as \(c^4\) or \(c^2d^2\).

By combining these components, we form algebraic expressions like \(c^4-c^2d^2+10c^2-6d^2+23\). Evaluating and manipulating these expressions require understanding the properties of operations involved.

Long Division

Long division is a technique used to divide complex expressions, much like the method used in basic arithmetic. It's especially helpful when dividing polynomials. This involves several steps to systematically reduce the expression towards the quotient and remainder.
To perform long division on polynomials, follow these steps:

• Divide the Leading Terms: Start by dividing the first term of the dividend by the first term of the divisor. This gives the first term of the quotient.
• Multiply: Multiply the entire divisor by this new term from the quotient.
• Subtract: Subtract the result from the original dividend to form a new dividend.
• Repeat Steps: Continue dividing until the degree of the new dividend is less than the degree of the divisor.

In practice, this method helps break down complex expressions into manageable parts, as seen when dividing \(c^4-c^2d^2+10c^2-6d^2+23\) by \(c^2 + 6\).

Dividing Polynomials

Dividing polynomials involves finding how many times a polynomial can be contained in another. This is akin to dividing numbers but focuses on expressions with variables. It's crucial to check the degree of polynomials during this process.

• Identify the Polynomial Parts: Before beginning the division, identify which polynomial is the dividend (what is divided) and which is the divisor (what you're dividing by).
• Determine Degrees: The degree of a polynomial is the highest power of the variable. This helps in determining how to start the division.
• Seek Patterns: Each term in the division process is derived from systematically reducing each polynomial's degree.
• Find the Quotient and Remainder: The process involves creating a quotient that represents how many times the divisor fits into the dividend completely, plus any remainder left over.

Applying this method, as in the example, results in the quotient \(c^2 - d^2 + 4\) with a remainder of \(-1\), showing all parts align with polynomial division principles.