Problem 45 - Algebra 1: Concepts and Skills

Question

Write the quotient in simplest form. $$\frac{4 x+3}{x-1} \div\left(4 x^{2}+x-3\right)$$

Step-by-Step Solution

Answer

The result of the simplification is \(\frac{1}{(x-1)(x+1)}\).

Step 1: Rewrite the Division as Multiplication

The expression can be rewritten as follows using the fact that \(a \,/\, b = a \cdot (1/b)\): \[ \frac{4x+3}{x-1} \cdot \frac{1}{4x^2+x-3} \]

Step 2: Factorize the Polynomial

Now, we need to factorize the quadratic polynomial \(4x^2+x-3\) to simplify the expression. By factoring the polynomial, we get \( (4x-3)(x+1) \). The expression can be rewritten as: \[ \frac{4x+3}{x-1} * \frac{1}{(4x-3)(x+1)} \]

Step 3: Simplify the Expression

Finally, you can simplify the expression and cancel out common terms. In this case, the terms \(4x+3\) and \(4x-3\) can be reduced, leading to a simplified result: \[ \frac{1}{(x-1)(x+1)} \]

Key Concepts

Division of Polynomials

When we talk about the division of polynomials, we are referring to the process of dividing one polynomial by another. The division of polynomials is similar to the division you would perform with numbers, but it involves finding a quotient and remainder that are also polynomials. To divide polynomials effectively, you can use long division or synthetic division methods.

To transform a division problem into a multiplication problem, you can employ the reciprocal of the divisor. This technique is particularly useful when working with algebraic fractions, like in our example, \( \frac{4x+3}{x-1} \div (4x^2+x-3) \). The division sign is swapped for multiplication, and we multiply by the reciprocal of the second polynomial. Understanding how to handle this flip from division to multiplication of polynomials is crucial for simplifying algebraic fractions.

Factoring Polynomials

Factoring polynomials is a key skill in algebra that involves breaking down a polynomial into its component factors, such that when multiplied together, they give back the original polynomial. This can simplify the multiplication and division of rational expressions. In our example, the polynomial \(4x^2+x-3\) was factored into \(4x-3\) and \(x+1\).

To factor a polynomial, one must look for common factors in the terms and apply various factoring techniques like finding the greatest common factor (GCF), using the difference of squares, or employing the quadratic formula for more complex polynomials. Factoring is essential before simplifying because it can reveal common terms in the numerator and denominator that may cancel out allowing for further simplification.

Multiplication and Division of Rational Expressions

Rational expressions are fractions that contain polynomials in both the numerator and the denominator. Multiplication and division of rational expressions follow the same basic principles as multiplying and dividing numeric fractions. To multiply rational expressions, you multiply the numerators with each other and the denominators with each other. Division, on the other hand, requires you to multiply by the reciprocal of the divisor.

When simplifying before multiplying or dividing, you should factor the polynomials in both the numerator and the denominator to identify and cancel out common factors. This simplification step is crucial as it makes the multiplication or division process much easier and the resulting expression much simpler, as demonstrated in our exercise where the common factors were identified and canceled to simplify the expression \( \frac{4x+3}{x-1} \cdot \frac{1}{4x^2+x-3} \).