Verified Answer :

  • Distribute the terms of the first binomial over the second using the distributive property: \(x(3x) + x(7) + (-5)(3x) + (-5)(7)\).
  • Calculate the products: \(3x^2 + 7x - 15x - 35\).
  • Combine the like terms \(7x\) and \(-15x\) to get \(-8x\).
  • The final simplified expression is \(\boxed{3x^2 - 8x - 35}\).

Explanation

  1. Applying the Distributive Property

To expand the expression \((x-5)(3x+7)\), we use the distributive property, which states that for any expressions \(a, b, c,\) and \(d\), the product \((a+b)(c+d)\) can be expanded as \(a \cdot c + a \cdot d + b \cdot c + b \cdot d\). In our case, we have \(a = x, b = -5, c = 3x,\) and \(d = 7\). This process is often called the FOIL method, where we multiply the First, Outer, Inner, and Last terms.

  1. Expanding the Binomials

Now, let's apply the multiplication to each pair of terms:

  • The First terms: \(x \times 3x = 3x^2\)
  • The Outer terms: \(x \times 7 = 7x\)
  • The Inner terms: \(-5 \times 3x = -15x\)
  • The Last terms: \(-5 \times 7 = -35\)

Adding these products together, we get: \(3x^2 + 7x - 15x - 35\).

  1. Simplifying the Expression

Finally, we combine the like terms, which are \(7x\) and \(-15x\). Adding these coefficients gives \(7x - 15x = -8x\). Thus, the fully expanded and simplified expression is \(3x^2 - 8x - 35\).

Examples

Expanding binomials is a fundamental skill in algebra with many real-world applications. For example, if you are designing a rectangular garden and know its length is represented by \((x+7)\) and its width by \((x-5)\), the area of the garden is the product of these two expressions. Calculating the total area as \(x^2 + 2x - 35\) allows you to determine how much space you have for plants or how much fencing you need based on the value of \(x\). Mastery of this technique provides the foundation for solving more complex quadratic equations in fields like engineering, economics, and physics.

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