Which of the following are the partial products for $\left(3 c^2+2 d\right)\left(-5 c^2+d\right)$? Choose four correct answers.
To find the partial products of the expression \((3c^2 + 2d)(-5c^2 + d)\), we utilize the distributive property, commonly referred to as the FOIL method for binomial multiplication. The FOIL method allows us to multiply each term in the first binomial by each term in the second binomial. The formula for the expansion of \((a + b)(x + y)\) is given by \(ax + ay + bx + by\). In our specific expression, we have \(a = 3c^2\), \(b = 2d\), \(x = -5c^2\), and \(y = d\).
Applying the FOIL steps systematically:
Comparing our results \(-15c^4\), \(3c^2d\), \(-10c^2d\), and \(2d^2\) with the provided options, we find that these four values match exactly with four of the choices given. Therefore, the correct partial products are \(-15c^4\), \(3c^2d\), \(-10c^2d\), and \(2d^2\).
In real-world engineering or physics, such as calculating the area of a rectangle whose dimensions are changing, you might encounter expressions involving variables like \((3w^2 + 2h)(-5w^2 + h)\). Expanding these expressions is a fundamental skill for simplifying physical models, determining net forces, or calculating areas and volumes in geometry. By breaking down the product into its partial components, you can more easily analyze how each variable contributes to the final outcome.