- Group the terms of the polynomial \(2x^4 + 5x^3 - 8x - 20\) into two pairs: \((2x^4 + 5x^3)\) and \((-8x - 20)\).
- Factor out the common term \(x^3\) from the first pair to get \(x^3(2x + 5)\).
- Factor out \(-4\) from the second pair to get \(-4(2x + 5)\).
- Combine the expressions to obtain \(x^3(2x + 5) - 4(2x + 5)\), which corresponds to option d.
- Final Answer: \(\boxed{d}\)
Explanation
- Problem Analysis
The objective is to find which expression from the given options is equivalent to the polynomial \(p(x) = 2x^4 + 5x^3 - 8x - 20\). To do this, we can use the technique of factoring by grouping, which involves splitting the polynomial into smaller groups of terms, factoring out common terms from each group, and then identifying a common binomial factor.
- Factoring by Grouping
We start by grouping the four terms of the polynomial into two pairs:
\((2x^4 + 5x^3)\) and \((-8x - 20)\).
Next, we factor out the greatest common factor from each group.
For the first pair, \((2x^4 + 5x^3)\), the common factor is \(x^3\), which gives:
\(x^3(2x + 5)\).
For the second pair, \((-8x - 20)\), we can factor out \(-4\), which gives:
\(-4(2x + 5)\).
- Identifying the Correct Expression
Combining the factored parts, we get:
\(x^3(2x + 5) - 4(2x + 5)\).
Now, we compare this expression to the given choices:
A\(x^3(2x - 5) - 4(2x - 5)\)
B\(x^3(2x + 5) + 4(2x + 5)\)
C\(x^3(2x + 5) + 4(2x - 5)\)
D\(x^3(2x + 5) - 4(2x + 5)\)
The resulting expression matches option (d). Therefore, the equivalent expression is \(x^3(2x + 5) - 4(2x + 5)\).
Examples
Factoring by grouping is a fundamental tool in algebra used to simplify complex polynomial expressions. In engineering, it can be used to solve for roots of structural equations, such as determining the stability points of a beam or calculating the stress distribution across a material. By reducing a higher-degree polynomial into a product of smaller factors, we can more easily identify critical values and understand the underlying dynamics of the system. For example, if you are modeling the trajectory of an object, breaking down the equation into simpler factors allows you to quickly find the times at which the object hits the ground or reaches a certain height.