Chapter 11: Problem 24
Challenge Problems $$\frac{3 a^{2}-4 a b+b^{2}}{a^{2}-a b}$$
Short Answer
Expert verified
\(\frac{3a-b}{a}\)
Step by step solution
01
Factor both the numerator and the denominator
Look for common factors in both the numerator and the denominator. The numerator is a trinomial that resembles the form of a square of a binomial, and the denominator is a binomial that can be factored by common terms. The factored forms are: Numerator: \(3a^2-4ab+b^2 = (a-b)(3a-b)\), Denominator: \(a^2-ab = a(a-b)\).
02
Reduce the common factors
Divide out the common factor of \(a-b\) that appears in both the numerator and the denominator. \[\frac{(a-b)(3a-b)}{a(a-b)} = \frac{3a-b}{a}\]
03
Simplify the result
The reduced form \(\frac{3a-b}{a}\) is already simplified. This is the most simplified form of the given expression.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Understanding how to factor polynomials is a fundamental skill in algebra that allows us to simplify complex expressions and solve equations. Factoring involves breaking down a polynomial into simpler terms, called factors, that when multiplied, give back the original polynomial.
For example, when you see a polynomial like \(3a^2 - 4ab + b^2\), recognizing patterns can help you factor it efficiently. This particular polynomial is known as a perfect square trinomial because it fits the form \(a^2 - 2ab + b^2\), which factors into \( (a-b)^2 \). However, it can also be factored as \(a-b)(3a-b)\), which is done by looking for a common factor in the terms, which in this case is less obvious but more suitable for simplification when the expression is part of a fraction.
To master factoring, practice identifying common factors and recognizing special forms like the difference of squares, perfect square trinomials, and the sum and difference of cubes. Using these strategies, you can transform seemingly intricate polynomials into products of binomials or monomials, paving the way for simplification and easier calculation.
For example, when you see a polynomial like \(3a^2 - 4ab + b^2\), recognizing patterns can help you factor it efficiently. This particular polynomial is known as a perfect square trinomial because it fits the form \(a^2 - 2ab + b^2\), which factors into \( (a-b)^2 \). However, it can also be factored as \(a-b)(3a-b)\), which is done by looking for a common factor in the terms, which in this case is less obvious but more suitable for simplification when the expression is part of a fraction.
To master factoring, practice identifying common factors and recognizing special forms like the difference of squares, perfect square trinomials, and the sum and difference of cubes. Using these strategies, you can transform seemingly intricate polynomials into products of binomials or monomials, paving the way for simplification and easier calculation.
Reducing Algebraic Expressions
The goal of reducing algebraic expressions is to present them in their most compact and understandable form. Reduction often involves factoring, like what we discussed in the context of polynomials, followed by canceling out common factors between the numerator and denominator of a fraction.
In our exercise, after factoring, we identified a common binomial \(a-b\) in both the numerator and the denominator. By canceling this common factor, we simplified the expression to \(\frac{3a-b}{a}\). The process of canceling is justified because dividing anything by itself, except for zero, results in one. Therefore, when we see the same factor in the top and bottom of a fraction, we can remove it from both, effectively 'reducing' the expression.
Always remember to look for opportunities to take out common factors when reducing expressions. It is also important to be vigilant about any restrictions on the variable, such as values for which the denominator would be zero, as these might become excluded from the domain of the expression. Reducing expressions correctly can lead to simpler solutions and a better understanding of algebraic relationships.
In our exercise, after factoring, we identified a common binomial \(a-b\) in both the numerator and the denominator. By canceling this common factor, we simplified the expression to \(\frac{3a-b}{a}\). The process of canceling is justified because dividing anything by itself, except for zero, results in one. Therefore, when we see the same factor in the top and bottom of a fraction, we can remove it from both, effectively 'reducing' the expression.
Always remember to look for opportunities to take out common factors when reducing expressions. It is also important to be vigilant about any restrictions on the variable, such as values for which the denominator would be zero, as these might become excluded from the domain of the expression. Reducing expressions correctly can lead to simpler solutions and a better understanding of algebraic relationships.
Binomial Expressions
A binomial expression is an algebraic expression that contains two terms, such as \(a-b\) or \(3a-b\). These are the building blocks for various algebraic operations, including the factoring process we utilize in simplifying algebraic fractions.
In the given exercise, the binomial \(a-b\) occurs as a common factor in both the numerator and denominator, making it a key player in the simplification process. By recognizing this, we efficiently reduced the fraction to its most simplified form. Binomial expressions often appear in factored forms of polynomials and can also be manipulated using arithmetic operations: addition, subtraction, multiplication, and division.
Working with binomial expressions involves understanding how to apply the distributive property, factoring techniques, and special binomial products like the square of a binomial and the product of a sum and difference. Developing a strong grasp of binomials paves the way for tackling more complex algebraic expressions and understanding the relationships between different algebraic concepts.
In the given exercise, the binomial \(a-b\) occurs as a common factor in both the numerator and denominator, making it a key player in the simplification process. By recognizing this, we efficiently reduced the fraction to its most simplified form. Binomial expressions often appear in factored forms of polynomials and can also be manipulated using arithmetic operations: addition, subtraction, multiplication, and division.
Working with binomial expressions involves understanding how to apply the distributive property, factoring techniques, and special binomial products like the square of a binomial and the product of a sum and difference. Developing a strong grasp of binomials paves the way for tackling more complex algebraic expressions and understanding the relationships between different algebraic concepts.
