Verified Answer :
- Identify the expression to expand: \((2x + 3)(x - 5)\).
- Apply the distributive property to multiply each term in the first binomial by each term in the second: \(2x(x) + 2x(-5) + 3(x) + 3(-5)\).
- Simplify the resulting terms: \(2x^2 - 10x + 3x - 15\).
- Combine the like terms \(-10x\) and \(3x\) to get the final result: \(\boxed{{2x^2 - 7x - 15}}\).
Explanation
- Problem Analysis
To expand the product of two binomials \((2x + 3)(x - 5)\), we use the distributive property, also known as the FOIL method (First, Outer, Inner, Last) or double distribution. This method ensures that each term in the first binomial is multiplied by every term in the second binomial. The objective is to convert the product into a standard quadratic polynomial in the form \(ax^2 + bx + c\).
- Expansion Using Double Distribution
We multiply each term of the first binomial \((2x + 3)\) by each term of the second binomial \((x - 5)\):
- Multiply the first terms: \(2x \times x = 2x^2\)
- Multiply the outer terms: \(2x \times (-5) = -10x\)
- Multiply the inner terms: \(3 \times x = 3x\)
- Multiply the last terms: \(3 \times (-5) = -15\)
By summing these products, we get the expanded expression: \(2x^2 - 10x + 3x - 15\).
- Combining Like Terms and Final Result
After expanding, we look for like terms to simplify the expression. The terms \(-10x\) and \(3x\) are both linear terms (terms containing \(x\)). We combine them:
\(-10x + 3x = -7x\)
The quadratic term \(2x^2\) and the constant term \(-15\) have no other like terms to combine with. Therefore, the fully simplified polynomial is \(2x^2 - 7x - 15\).
Examples
Expanding binomials is useful in fields like geometry and engineering. For example, if you are designing a rectangular garden where the length is \((2x + 3)\) meters and the width is \((x - 5)\) meters, the total area of the garden is the product of its dimensions: \((2x + 3)(x - 5)\). By expanding this expression, you obtain the formula \(2x^2 - 7x - 15\) for the area in terms of \(x\), which allows you to calculate the necessary resources, like soil or fencing, for any given value of \(x\).