Problem 39 - Elementary and Intermediate Algebra

Question

Perform each division. See Examples 3 and \(4 .\) Divide \(x^{2}-5 x+6\) by \(x-3\)

Step-by-Step Solution

Answer

The quotient is \(x - 2\).

Step 1: Set Up the Division

We will use polynomial long division to divide the polynomial \(x^2 - 5x + 6\) by \(x - 3\). Start by setting up the division, placing \(x^2 - 5x + 6\) under the division symbol and \(x - 3\) outside.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \(x^2\), by the leading term of the divisor, \(x\). This gives \(x^2 \div x = x\). Write \(x\) above the division line as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(x - 3\) by the term \(x\) just obtained, which gives \(x(x - 3) = x^2 - 3x\). Subtract this result from the original dividend \(x^2 - 5x + 6\), resulting in a new polynomial: \((x^2 - 5x + 6) - (x^2 - 3x) = -2x + 6\).

Step 4: Repeat the Process

Divide the new leading term \(-2x\) by \(x\) to get \(-2x \div x = -2\). Write \(-2\) as the next term of the quotient. Multiply \(-2\) by \(x - 3\), giving \(-2(x - 3) = -2x + 6\).

Step 5: Subtract Again

Subtract \(-2x + 6\) from \(-2x + 6\) which gives \((-2x + 6) - (-2x + 6) = 0\). There is no remainder.

Step 6: Write the Final Quotient

Since the remainder is \(0\), the division is exact and the quotient is \(x - 2\).

Key Concepts

Long Division Method

The Long Division Method is a powerful technique used to divide polynomials, much like how you might divide large numbers. This method helps simplify complex equations and find the exact quotient when dividing. The process is straightforward, involving several key steps.

• First, set up the division by placing the dividend (the polynomial you're dividing) under a division symbol and the divisor (the polynomial you're dividing by) outside.
• Next, divide the first term of the dividend by the first term of the divisor. This initial division helps you establish the first term of the quotient.
• Then, multiply the entire divisor by this new term from the quotient, and subtract the result from the dividend.
• This subtraction generates a new polynomial, and you repeat the process with this new polynomial until no terms are left.

Breaking it down into smaller pieces, the Long Division Method ensures clarity and avoids errors. It's invaluable in algebra where precision and accuracy are necessary.

Dividing Polynomials

Dividing Polynomials can seem tricky at first, but with practice, it becomes quite systematic. The main objective is to simplify a complex equation by dividing it into smaller, manageable terms. This allows you to reach a more useful and workable solution.

• Start with the leading term of the dividend and the leading term of the divisor. This is your strategic move to get a clear start.
• With each division and subtraction, as you work your way through the polynomial, you learn more about the structure of the equation.
• The process brings a satisfaction of tidying up a complex expression into a neat equation, often referred to as the quotient.

Understanding how to divide polynomials is crucial in solving many algebraic equations as it forms the foundation of more advanced algebraic techniques.

Algebraic Equations

Algebraic Equations are mathematical statements indicating the equality of two expressions. Dividing polynomials using the long division method often arises when solving these equations. Simply put, it involves determining the quotient which satisfies the division.

• Many algebraic equations involve polynomials that are divided to find values that satisfy the equation.
• These operations might appear daunting but using systematic methods such as polynomial division makes the equations more approachable.
• Solving algebraic equations is essential in many aspects of mathematics. It helps to resolve real-world problems by providing solutions through logical sequences.

In the context of polynomial division, you're not just seeking a quotient; you're finding an expression that reveals more about the relationship between the variables involved in your equation.