Problem 12 - Algebra for College Students

Question

Perform the indicated divisions. $$ \frac{x^{2}+11 x-60}{x-4} $$

Step-by-Step Solution

Answer

The quotient is \(x + 15\).

Step 1: Set Up the Problem

First, identify the given polynomial division problem. You need to divide the polynomial \(x^2 + 11x - 60\) by \(x - 4\). This problem requires using polynomial long division.

Step 2: Start the Division

Begin by dividing the first term of the dividend \(x^2\) by the first term of the divisor \(x\) to get \(x\). Write \(x\) above the division line. This is the first term of the quotient.

Step 3: Multiply and Subtract

Next, multiply \(x\) by \(x - 4\), getting \(x^2 - 4x\). Subtract this from the original dividend \(x^2 + 11x - 60\), which gives you a new polynomial: \(15x - 60\).

Step 4: Repeat the Process

Now divide the first term of the new polynomial \(15x\) by the first term of the divisor \(x\) to get \(15\). Write \(15\) in the quotient next to \(x\). Then multiply \(15\) by \(x - 4\) giving you \(15x - 60\). Subtract \(15x - 60\) from \(15x - 60\), which leaves you with \(0\).

Step 5: Write the Solution

Since the remainder is \(0\), the division is exact. The quotient is the result of the division process, which is \(x + 15\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method similar to the long division you learned in arithmetic, but it's applied to polynomials. This technique is essential for dividing one polynomial by another, especially when dealing with higher degree polynomials.

The process involves dividing the larger polynomial, known as the dividend, by a smaller polynomial called the divisor. In this case, we want to divide the polynomial \(x^2 + 11x - 60\) by \(x - 4\). Here are the steps involved:

• Identify the dividend and the divisor.
• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract the resulting polynomial from the dividend to find a new polynomial.
• Repeat the process with the new polynomial until the degree of the remaining polynomial is less than the degree of the divisor.

The result of polynomial long division provides us with both a quotient and, occasionally, a remainder, showing us how the dividend can be expressed as a combination of the divisor and the additional polynomial terms. This method is an invaluable skill in algebra for simplifying complex polynomial expressions.

Dividing Polynomials

Dividing polynomials involves reducing a complex polynomial expression into smaller, more manageable parts. This is done through operations like polynomial long division or synthetic division. When you divide polynomials, you're essentially splitting them into parts and understanding how one polynomial behaves with respect to another.

To effectively divide polynomials:

• Always start by arranging the polynomials in descending order of power.
• Focus on dividing the leading term to simplify the process.
• Carry out subtraction carefully, paying close attention to each term.
• Continue the division process repetitively for accuracy.

When dividing \(x^2 + 11x - 60\) by \(x - 4\), each step is crucial for deriving the correct quotient, \(x + 15\), and helps in understanding the intrinsic relationship between the two polynomials. Mastering polynomial division simplifies many algebraic processes and aids in solving more complex equations.

Remainder Theorem

The Remainder Theorem provides a quick way to find the remainder of a polynomial division. According to the theorem, if you divide a polynomial \(f(x)\) by a simple factor \((x - c)\), the remainder of this division is simply \(f(c)\).

In practical terms, this means you only need to substitute the value of \(c\) into the polynomial. If \(x - 4\) is the divisor, evaluate \(f(4)\). For our division, evaluating \(x^2 + 11x - 60\) at \(x = 4\) results in a remainder of 0. This confirms the division is exact, and the divisor \(x - 4\) is a factor of the polynomial \(x^2 + 11x - 60\).

The Remainder Theorem is a valuable tool for verifying the results of polynomial division, confirming factors, and simplifying complex algebraic tasks without extensive calculation. When there is no remainder, as in our case, the quotient obtained from the division is the complete solution.

Quotients in Algebra

Quotients are the results you get when you divide one quantity by another. In algebra, specifically when working with polynomials, the quotient is the polynomial part of the division that doesn’t include the remainder.

In our exercise, after dividing \(x^2 + 11x - 60\) by \(x - 4\), the quotient is \(x + 15\). This means the original polynomial can be expressed as \((x-4)(x+15)\) when expanded back.

• The quotient helps in simplifying expressions and solving equations.
• Understanding the role of the quotient is crucial for factoring polynomials.
• Quotients are often used in calculus and higher algebra when integrating or differentiating polynomial functions.

Knowing how to find and interpret quotients helps in deepening your understanding of polynomial relationships and is pivotal in many areas of mathematical study.