Problem 93 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{t^{3}-5 t^{2}+6 t}{9 t-t^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{t-2}{t+3}\).

Step 1: Factor the numerator

The numerator of the expression is \( t^3 - 5t^2 + 6t \). Notice that each term contains a factor of \( t \). Thus, factor \( t \) out of the entire expression, yielding \( t(t^2 - 5t + 6) \). Next, factor the quadratic expression \( t^2 - 5t + 6 \). The quadratic can be factored as \( (t - 2)(t - 3) \). Hence, the fully factored form of the numerator is \( t(t-2)(t-3) \).

Step 2: Factor the denominator

The denominator of the expression is \( 9t - t^3 \). Rearrange it as \( -t^3 + 9t \). Factor out the common factor of \( t \) to obtain \( t(-t^2 + 9) \). Notice that \( -t^2 + 9 \) is a difference of squares, and it can be expressed as \( -(t^2 - 9) \), which factors into \( -(t - 3)(t + 3) \). Thus, the fully factored denominator is \( -t(t - 3)(t + 3) \).

Step 3: Cancel common factors

Write the full expression with the factored forms: \( \frac{t(t-2)(t-3)}{-t(t-3)(t+3)} \). Identify the common factors in the numerator and the denominator, which in this case are \( t \) and \( t - 3 \). Cancel these common factors: \( \frac{(t-2)}{-(t+3)} \). Simplify the expression to \( \frac{t-2}{-(t+3)} \).

Step 4: Simplify the expression

Rewrite \( \frac{t-2}{-(t+3)} \) as \( -\frac{t-2}{t+3} \) by factoring out the negative sign. This is the simplified form of the expression.

Key Concepts

Factoring Quadratics

Factoring quadratics is like trying to find pieces of a puzzle that fit perfectly into the bigger picture. When you see an expression like \( t^2 - 5t + 6 \), your goal is to break it down into simpler expressions that, when multiplied together, regenerate the original quadratic.

To factor a quadratic, you look for two numbers that both multiply to give the last term (constant term) and add to give the middle term. In our example, we need two numbers that multiply to 6 and add to -5. The numbers -2 and -3 fit perfectly! Thus, the expression factors into \((t-2)(t-3)\).

This way of simplifying expressions is key in algebra because it turns complex quadratic equations into simpler, solvable forms. Practice spotting those number pairs, and soon, factoring quadratics will become second nature.

Rational Expressions

Rational expressions are fractions, but instead of having single numbers on top and bottom, they have polynomials. Simplifying rational expressions is about making these fractions as simple as possible.

Picture the original expression \( \frac{t^3 - 5t^2 + 6t}{9t - t^3} \) as a fraction where both the numerator and denominator can be factored to reveal common parts. By factoring these parts, we can "cancel" them out, much like you might reduce the fraction \( \frac{4}{8} \) by recognizing both numbers are divisible by 4.

The process involves:

• Factoring the numerator and the denominator separately.


• Identifying common factors in both.


• Cancelling those common factors out to simplify the expression.

Mastering rational expressions allows you to simplify and solve more complex algebraic equations with ease.

Difference of Squares

The difference of squares feels almost magical because it's a special case that makes factoring much simpler. When you see an algebraic expression like \( a^2 - b^2 \), you can instantly factor it into \((a-b)(a+b)\). This pattern holds true for any values of \(a\) and \(b\).

In our exercise, the term \(-t^2 + 9\) is a classic example of a difference of squares. Here, it can be rewritten as \(-(t^2 - 9)\) which factors to \(-(t-3)(t+3)\). Recognizing this pattern can save you a lot of time and effort while working through algebra problems.

This concept is crucial because it often appears in integrals, derivatives, and more advanced algebra. Once you grasp the difference of squares, you can tackle many mathematical challenges more confidently.