- Expand the product \((3c^2+2d)(-5c^2+d)\) using the FOIL method.
- Calculate the products of the First (\(-15c^4\)), Outer (\(3c^2d\)), Inner (\(-10c^2d\)), and Last (\(2d^2\)) terms.
- Combine the like terms \(3c^2d\) and \(-10c^2d\) to get \(-7c^2d\).
- The simplified expression is \(-15c^4-7c^2d+2d^2\), which corresponds to option a. The final answer is \(\boxed{a}\).
Explanation
- Problem Analysis
The objective is to expand the product of two binomials: \((3c^2 + 2d)\) and \((-5c^2 + d)\). This is a standard algebraic multiplication problem that can be solved using the distributive property, often referred to as the FOIL (First, Outer, Inner, Last) method. The expression is given as:
\[(3c^2 + 2d)(-5c^2 + d)\]
Our goal is to multiply these two terms together and simplify the resulting expression to match one of the given choices.
- Applying the Distributive Property
To expand the expression \((3c^2 + 2d)(-5c^2 + d)\), we multiply each term in the first binomial by each term in the second binomial:
- First: Multiply the first terms of each binomial:
\[(3c^2) \times (-5c^2) = -15c^4\]
- Outer: Multiply the outer terms of the product:
\[(3c^2) \times (d) = 3c^2d\]
- Inner: Multiply the inner terms of the product:
\[(2d) \times (-5c^2) = -10c^2d\]
- Last: Multiply the last terms of each binomial:
\[(2d) \times (d) = 2d^2\]
- Combining Like Terms and Conclusion
Now, we sum all the results obtained from the distribution step to form the expanded polynomial:
\[-15c^4 + 3c^2d - 10c^2d + 2d^2\]
To simplify this expression, we combine the like terms \(3c^2d\) and \(-10c^2d\):
\[3c^2d - 10c^2d = (3 - 10)c^2d = -7c^2d\]
Substituting this back into the expression, we get:
\[-15c^4 - 7c^2d + 2d^2\]
Comparing this result with the given options, we see that it matches option (a).
Examples
In many engineering and physics applications, we often need to calculate areas or work done where variables involve powers, such as rectangles with dimensions \(3c^2 + 2d\) and \(-5c^2 + d\) (though dimensions are typically positive, algebraic expansion remains a crucial tool). Expanding expressions is fundamental in simplifying complex formulas used in structural analysis or electrical circuit design, where terms like these appear when analyzing voltage or stress distributions over variable surfaces.