Question

Simplify \(\frac{\left(4 x^{2}-7\right) \cdot 3-(3 x+5) \cdot 8 x}{\left(4 x^{2}-7\right)^{2}}\)

Short answer

\( \frac{-12x^2 - 40x - 21}{(4x^2 - 7)^2} \)

Step-by-step solution

- Distribute terms in the numerator

First, distribute the constants 3 and 8x across their respective terms in the numerator. This gives us: \[ 3(4x^2 - 7) - 8x(3x + 5) \]

- Simplify each distribution

Next, we simplify the distributed terms: 3(4x^2) + 3(-7) - 8x(3x) - 8x(5) This results in: \[ 12x^2 - 21 - 24x^2 - 40x \]

- Combine like terms in the numerator

Combine the like terms in the numerator: \[ 12x^2 - 24x^2 - 40x - 21 \] This results in: \[ -12x^2 - 40x - 21 \]

- Combine the simplified numerator with the denominator

Write the simplified numerator over the original denominator: \[ \frac{ -12x^2 - 40x - 21 }{(4x^2 - 7)^2} \]

Key concepts

Distributing Terms

Let's start by understanding the concept of distributing terms. When you distribute, you are essentially breaking down an expression to make it simpler. Think of it as multiplying each term inside a parenthesis by the number outside. For example, in the expression \(3(4x^2 - 7)\), you will multiply 3 with each term inside the parenthesis. This gives: \(3 \times 4x^2 + 3 \times (-7)\), resulting in \(12x^2 - 21\).
Similarly, in \((-8x)(3x + 5)\), you distribute \(-8x\) across \((3x + 5)\): \(-8x \times 3x + (-8x) \times 5\), which results in \(-24x^2 - 40x\).
By understanding distribution, you can tackle more complex polynomial expressions more easily.

Combining Like Terms

Combining like terms helps in simplifying polynomial expressions. Like terms are terms that contain the same variable raised to the same power. For example, \(4x^2\) and \(12x^2\) are like terms because both contain \(x^2\).
In the expression \(12x^2 - 21 - 24x^2 - 40x\), the like terms are \(12x^2\) and \(-24x^2\). Combining these gives: \[12x^2 - 24x^2 = -12x^2.\]
The term \(-21\) is a constant, and \(-40x\) is a linear term; both are not like terms with \(-12x^2\). Therefore, the simplified expression after combining like terms becomes: \(-12x^2 - 40x - 21\).
Keep in mind that combining like terms reduces the expression into a simpler form where fewer terms are present.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator and the denominator. In our exercise, we have the rational expression: \[\frac{ -12x^2 - 40x - 21 }{(4x^2 - 7)^2}\]
Rational expressions can be simplified by performing operations like simplifying the numerator and denominator separately, and then reducing any common factors between them. In our case, by simplifying the numerator first and then writing it over the original denominator, we get a simplified rational expression. Simplifying both components of the fraction allows for a cleaner, more understandable form.
Learning to handle rational expressions properly helps to solve many algebraic problems more efficiently and accurately.