Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Divide using long division. $$\left(x^{2}+5 x+6\right) \div(x-4)$$

Step-by-Step Solution

Answer

The result of the division \(\frac{x^{2}+5x+6}{x-4}\) is \(x + 9 + \frac{42}{x - 4}\)

Step 1: Setting Up the Long Division

We write the division problem in long division format. In the place of the dividend (inside the right part of the symbol), we have the function \(x^{2} + 5x + 6\), and the divisor (on the left of the symbol), we place the function \(x - 4\).

Step 2: Division

Here we divide the first term of the dividend, \(x^{2}\), by the first term of the divisor, \(x\), to get \(x\). We then multiply the whole divisor, \(x - 4\), by \(x\) and we subtract the result from the original polynomial. This gives us a new polynomial, \(9x + 6\).

Step 3: Division Again

We divide the first term of the new polynomial, \(9x\), by the first term of the divisor again, \(x\), to get \(9\). We then multiply the whole divisor, \(x - 4\), by \(9\) and subtract the result again from the new polynomial. This gives us a remainder of \(42\).

Step 4: Write the Final Answer

We write the solution in the form of 'quotient + remainder/divisor', which is \(x + 9 + \frac{42}{x - 4}\)

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. This process is similar to long division with numbers but involves algebraic expressions. In this exercise, we use a method called 'long division' to solve the equation \( (x^2 + 5x + 6) \div (x - 4) \).

• First, we set up the division just like you would for numbers.
• We place the dividend \( (x^2 + 5x + 6) \) inside and the divisor \( (x - 4) \) outside.
• The goal is to find out how many times the divisor can fit into the dividend.

This iterative method helps in simplifying complicated algebraic expressions by breaking them down into simpler parts, facilitating understanding and computation.

Algebraic Expressions

Algebraic expressions form the backbone of polynomial division. They are mathematical phrases that can include numbers, variables, and operational symbols. In our exercise, \(x^2 + 5x + 6\) is the given algebraic expression or dividend, while \(x - 4\) serves as the divisor.

• Expression terms are crucial since they guide how division unfolds.
• In algebra, terms are separated by '+' or '-' signs.
• For example, in \(x^2 + 5x + 6\), the terms are \(x^2, 5x,\) and \(6\).

Understanding these components and how they interact with operations such as division helps you master complex and seemingly challenging algebraic tasks.

Remainder Theorem

The Remainder Theorem is a significant concept when dealing with polynomial divisions. It states that when a polynomial \(f(x)\) is divided by \(x - c\), the remainder is \(f(c)\). This means the remainder can directly inform us about the polynomial at a particular value of \(x\).

• In our case, \(x - 4\) is the divisor, suggesting we evaluate at \(x=4\).
• Calculating directly, \( f(4) = 4^2 + 5(4) + 6 = 42 \).
• This matches the remainder from our long division process, solidifying the concept.

Use of the Remainder Theorem simplifies the understanding of division outcomes, confirming results effectively.

Quotient and Remainder in Division

Dividing polynomials results in a quotient and possibly a remainder. The quotient is the result of the division, while any leftover part is known as the remainder.

• From the steps, the quotient is \(x + 9\), and the remainder is \(42\).
• The answer can be written as: \(x + 9 + \frac{42}{x - 4}\).
• This form \( \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}} \) is typical for polynomial division answers.

Understanding how to express the final result accurately in terms of quotient and remainder is essential for mastery in polynomial division exercises.