Problem 37 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{(3 t-1)^{2}}{45 t-15} \div \frac{12 t^{2}+5 t-3}{20 t+5}$$

Step-by-Step Solution

Answer

\(\frac{(3t-1)(4t+1)}{15(3t+1)(4t-3)}\) in its simplest form.

Step 1: Simplify the Division

Start by converting the division of two fractions into a multiplication by flipping the second fraction. We have: \[ \frac{(3t-1)^2}{45t-15} \times \frac{20t+5}{12t^2+5t-3}. \]

Step 2: Factor Each Expression

Factor the expressions if possible:1. \((3t-1)^2\) is already factored.2. \(45t - 15 = 15(3t - 1)\).3. \(20t + 5 = 5(4t + 1)\).4. For \(12t^2 + 5t - 3\), apply the quadratic formula factorization. - Find factors of \(-36\) (product of \(12\) and \(-3\)) that sum to \(5\): factors are \(9\) and \(-4\). - Rewrite and factor: \(12t^2 + 9t - 4t - 3 = (3t + 1)(4t - 3)\).

Step 3: Set Up the Factored Expression

After factoring, write the expression as:\[\frac{(3t-1)^2}{15(3t-1)} \times \frac{5(4t+1)}{(3t+1)(4t-3)}.\]

Step 4: Cancel Common Factors

Cancel out common factors in the numerator and the denominator:- Cancel \((3t-1)\) found in the numerator \((3t-1)^2\) and the denominator \(15(3t-1)\).- As a result, the expression simplifies to:\[ \frac{(3t-1)(4t+1)}{15(3t+1)(4t-3)}.\]

Step 5: Write the Final Simplified Expression

After simplification, the expression in its simplest form is:\[ \frac{(3t-1)(4t+1)}{15(3t+1)(4t-3)}. \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a key step to simplifying and solving polynomial equations. It involves breaking down a polynomial into simpler "factors" that, when multiplied together, give the original polynomial. Consider the expression

• \((3t-1)^2\)
• \(45t - 15\)
• \(12t^2 + 5t - 3\)

Each can be expressed as a product of simpler terms.
For example, \((3t-1)^2\) is already in its factored form. The expression \(45t - 15\) can be factored by taking out the greatest common factor, resulting in \(15(3t-1)\).
Polynomials can often be factored using methods such as grouping, using formulae like the quadratic formula, or by identifying special patterns such as perfect square trinomials or difference of squares. Here, \(12t^2 + 5t - 3\) is factored by looking for two numbers that multiply to \(-36\) and add to \(5\). This results in the factors \( (3t + 1)(4t - 3) \).
Being able to factor allows us to simplify expressions and solve equations more easily.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator, the denominator, or both. These expressions behave similarly to numerical fractions, where simplification or cancellation of terms is often necessary.
In tackling rational expressions like \(\frac{(3t-1)^2}{45t-15} \div \frac{12t^2+5t-3}{20t+5}\), it's essential first to convert division into multiplication by using the reciprocal of the second fraction:

• \(\frac{20t+5}{12t^2+5t-3}\)

Once converted, simplifying each expression involves factoring them to reveal common terms.
Simplifying rational expressions helps us not only in algebraic solutions but also in understanding the behavior of functions, particularly when these are applied to real-world problems.

Simplifying Expressions

Simplifying expressions entails reducing them to their most basic form, making them easier to understand and work with. This process often involves factoring and canceling common factors.
For instance, after factoring the polynomials of the expression \(\frac{(3t-1)^2}{15(3t-1)} \times \frac{5(4t+1)}{(3t+1)(4t-3)}\), we can cancel common terms across the numerator and the denominator. If the same polynomial appears both in the numerator and denominator, these can cancel out:

• The factor \((3t-1)\) in \((3t-1)^2\) cancels out one of the \((3t-1)\) in the denominator.

As a result, the expression simplifies to \(\frac{(3t-1)(4t+1)}{15(3t+1)(4t-3)}\), which is the simplest form. This final form is crucial in further computations or in drawing inferences in algebra.