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Chapter 4: Problem 31

Perform each division. Assume no division by \(0 .\) $$\frac{x^{2}-5 x+6}{x-2}$$

Short Answer

Expert verified
The result is \(x-3\).

Step by step solution

01

Factorization of the Numerator

The numerator is a quadratic equation that can be factorized. Let's factorize \(x^{2}-5x+6\). After factorization, it can be represented as \((x-2)(x-3)\)
02

Polynomial Division

We divide \(x^{2}-5x+6\) by \(x-2\) that equals \((x-2)(x-3) / (x-2)\). We have the common term \((x-2)\) in numerator and denominator, which can be cancelled out.
03

Simplify the Expression

After canceling out the common term, we are left with \(x-3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorization
Factorization is a crucial step when dealing with polynomials. It involves breaking down a complex expression into simpler, multipliable terms known as factors. In our example, we have a quadratic expression in the numerator:
  • \(x^{2}-5x+6\) can be broken into simpler expressions.
  • This process involves finding two numbers that multiply to 6 (the constant term) and add up to -5 (the linear coefficient).
Here, those numbers are -2 and -3. This means
  • \(x^{2}-5x+6\) can be factored into \((x-2)(x-3)\).
Once we've factorized, dividing becomes much easier. Factorization simplifies expressions by reducing them to their smallest parts, revealing patterns or similarities that can be simplified further.
Quadratic Equation
A quadratic equation is an equation in the form of \(ax^{2} + bx + c = 0\). The example provided, \(x^{2}-5x+6\), fits this format. Quadratic equations are usually solved by:
  • Factorization, when possible.
  • Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
  • Completing the square.
For our example, we opted for factorization because it quickly led to simplified terms. Factorization works well when dealing with simple quadratics where clear factors can be identified. Factoring not only simplifies solving but also understanding the roots of the equation. After factorization, it's easy to see the roots as \(x=2\) and \(x=3\), indicating where the polynomial hits zero.
Simplification
Simplification in polynomial division often involves canceling out identical terms in the numerator and denominator. In our exercise, after expressing the numerator as \((x-2)(x-3)\):
  • We place \((x-2)\) in the denominator as well.
  • Both the numerator and denominator share the factor \((x-2)\).
This allows us to cancel out \((x-2)\), leaving us with the simple expression \(x-3\). Simplification means reducing a complex expression into its simplest form to make it easier to understand or solve. It's a vital part of many algebraic processes and helps in quickly finding solutions or making calculations manageable.

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Most popular questions from this chapter

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