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Chapter 4: Problem 35

Perform each division. Assume no division by \(0 .\) $$\frac{8 a^{2}-10 a+3}{4 a-3}$$

Short Answer

Expert verified
The simplified expression is \((2a - 1)\).

Step by step solution

01

Factorize the Numerator

We rewrite \(8a^{2}-10a+3\) as \((4a - 1)(2a - 3)\). We can confirm that this is correct by multiplying the factors to get back the original expression.
02

Cancel Common Factors

We notice that the expression \((4a - 3)\) is a common factor in the numerator and denominator. Since we're dividing, we can cancel this factor out, leaving us with \((2a - 1)\) as the remaining expression.
03

Simplify the Resulting Expression

After canceling out the common factor, we're left with \((2a - 1)\). This is the simplified version of the original expression.

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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 skill in algebra often used to simplify expressions and solve equations.
  • In the given exercise, we factorize the polynomial in the numerator, which is a quadratic trinomial: \(8a^2-10a+3\).
  • To factorize this, we look for two binomials that multiply to give the original polynomial.
  • Factoring involves identifying two numbers that multiply to give the constant (here, 3) and add up to the middle coefficient (here, -10).
  • In our example, the trinomial \(8a^2-10a+3\) can be rewritten as \((4a - 1)(2a - 3)\).
These steps involve trial and error, inspection, or sometimes tools like the AC method. Once factored, check your work by expanding the factors to verify that they produce the original polynomial. This often involves distributing, combining like terms, and ensuring all terms return correctly.
Algebraic Expressions
Algebraic expressions consist of variables, numbers, and operations. They are the building blocks of algebra and need to be manipulated carefully.
  • The expression from the exercise, \(8a^2 - 10a + 3\), is an algebraic expression with terms in powers of \(a\).
  • Algebraic expressions can be combined, simplified, multiplied, and divided.
  • The division of one algebraic expression by another polynomial, like here with \(\frac{8a^2-10a+3}{4a-3}\), requires specific steps like factorization and cancellation.
When dealing with algebraic expressions, always keep the operations clear, understand the role of each component, and be careful not to divide by zero. Each operation follows specific rules that ensure expressions are maintained and transformed correctly during calculations.
Simplifying Expressions
Simplifying expressions is the process of reducing them to their most compact and manageable form. This makes it easier to solve equations or understand the expression itself.
  • Once the expression is factorized and any common factors are identified, the simplification involves canceling these common factors.
  • For our example, after rewriting the numerator as \((4a - 1)(2a - 3)\) and noting that \(4a-3\) also appears in the denominator, we can cancel this factor.
  • After canceling, you're left with \((2a-1)\), which is the simpler expression derived from the original.
Be mindful that not all expressions can be simplified in such a direct manner, but recognizing patterns and using factorization can help in many situations. Always reassess the final form to ensure that no further simplifications are possible.

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