Problem 110 - Intermediate Algebra

Question

Simplify each expression. $$ \frac{20 m^{2}\left(m^{2}-1\right)-47 m\left(1-m^{2}\right)+24\left(m^{2}-1\right)}{4 m^{2}-m-3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{20m^4 + 47m^3 + 4m^2 - 47m - 24}{(4m + 3)(m - 1)}\).

Step 1: Distribute Terms in Numerator

Distribute each term inside the parentheses for the expressions in the numerator: 1. For the first term: \(20m^2(m^2 - 1) = 20m^4 - 20m^2\).2. For the second term: \(-47m(1 - m^2) = -47m + 47m^3\).3. For the third term: \(24(m^2 - 1) = 24m^2 - 24\).

Step 2: Combine Like Terms in Numerator

Combine the like terms after distributing:1. The \(m^4\) term: \(20m^4\).2. The \(m^3\) term: \(47m^3\).3. The \(m^2\) terms: \(-20m^2 + 24m^2 = 4m^2\).4. The \(m\) term: \(-47m\).5. The constant term: \(-24\).Thus, the numerator becomes: \(20m^4 + 47m^3 + 4m^2 - 47m - 24\).

Step 3: Factor the Denominator

The denominator is \(4m^2 - m - 3\). Start by finding factors of the quadratic using the factorization method or the quadratic formula:The factors of \(4m^2 - m - 3\) are \((4m + 3)(m - 1)\).

Step 4: Simplify the Fraction

The expression is \(\frac{20m^4 + 47m^3 + 4m^2 - 47m - 24}{(4m + 3)(m - 1)}\).Since there are no common factors that can simplify both the numerator and denominator further, our simplified expression remains as it is.

Key Concepts

Distributing Terms

When simplifying rational expressions, one important step is distributing terms across the expression. This can be thought of as unwrapping the expression from its brackets. In essence, it's like multiplying each of the terms inside the brackets by the term outside it. This process allows us to expand expressions and makes it easier to see how the pieces fit together.

For example, in the expression from our problem, we have three products within the numerator. Distributing them means you'll multiply each inner term by the term outside the brackets:

• Start with the first part: Distribute the term \(20m^2\) over each part inside the parentheses: \(20m^2(m^2 - 1) = 20m^4 - 20m^2\).
• Next, take the second part: \(-47m(1 - m^2)\). Here, distribute \(-47m\): \(-47m + 47m^3\).
• Finally, the third part: Distribute \(24\) inside the brackets: \(24(m^2 - 1) = 24m^2 - 24\).

Through these steps, you'll end up with each component expanded and ready for the next steps of simplification.

Combining Like Terms

Once the terms are distributed, the next rational step is to combine like terms. This means grouping terms that have the same variable to the same power together. It's like consolidating your grocery list: instead of writing apples twice, combine them into one line with the total quantity.

In our exercise, after distributing the terms, we are left with multiple terms in the expanded numerator: \(20m^4 + 47m^3 + 4m^2 - 47m - 24\). Here is how you group them:

• The \(m^4\) term: Since there's only one, it remains \(20m^4\).
• The \(m^3\) term: Similarly, \(47m^3\) is by itself.
• For the \(m^2\) terms: Combine \(-20m^2\) and \(24m^2\) to get \(4m^2\).
• The \(m\) terms: There's just one, so it stays \(-47m\).
• And the constant term: \(-24\) stands alone.

Through these steps, the expression is more compact and set up for potential factoring or division.

Factoring Quadratics

Factoring quadratics is an essential skill in simplifying rational expressions. It's akin to finding out what makes a quadratic tick beneath its outer appearance. This process involves rewriting the quadratic as a product of simpler binomial expressions, similar to reverse engineering.

Focusing on the denominator of our expression, \(4m^2 - m - 3\), we need to find factors of this quadratic that multiply to give us the original expression. This can be done using several strategies:

• **The Factorization Method**: Look for two numbers that multiply to give the product of \(a\) and \(c\) (first and last coefficients) and add to \(b\) (middle coefficient), if possible. Here, it can be expressed as \((4m+3)(m-1)\).
• **Quadratic Formula**: Alternatively, use the formula \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) if the factorization isn't straightforward.

By factoring the quadratic, you break it down into its more basic components, making the entire expression easier to handle. It's this step that often allows for cancellation and further simplification in rational expressions, but always check if they share a common factor with the numerator to simplify further.