- Multiply the binomials \((3y - 4)\) and \((2y + 7)\) to get \(6y^2 + 21y - 8y - 28\).
- Simplify the product to \(6y^2 + 13y - 28\).
- Add the remaining terms \(11y - 9\) to the product.
- Combine all like terms to obtain the final simplified expression: \(\boxed{6y^2 + 24y - 37}\).
Explanation
- Problem Analysis
We need to simplify the expression \((3y - 4)(2y + 7) + 11y - 9\). The expression consists of a product of two binomials, \((3y - 4)\) and \((2y + 7)\), followed by the addition of the linear expression \(11y - 9\). Our strategy is to expand the binomial product using the distributive property, simplify it, and then combine the result with the remaining linear terms by grouping like terms.
- Expansion of the Binomial Product
First, we expand the binomial product \((3y - 4)(2y + 7)\) using the distributive property (often referred to as the FOIL method):
\[(3y - 4)(2y + 7) = (3y \cdot 2y) + (3y \cdot 7) + (-4 \cdot 2y) + (-4 \cdot 7)\]
Calculating each term:
- \(3y \cdot 2y = 6y^2\)
- \(3y \cdot 7 = 21y\)
- \(-4 \cdot 2y = -8y\)
- \(-4 \cdot 7 = -28\)
Combining these, we get:
\[6y^2 + 21y - 8y - 28 = 6y^2 + 13y - 28\]
This is the simplified result of the binomial product.
- Combining Like Terms and Final Simplification
Now, we add the remaining part of the original expression, \(11y - 9\), to our expanded result:
\[(6y^2 + 13y - 28) + 11y - 9\]
To find the final simplified expression, we group the like terms (the \(y\) terms and the constant terms):
- The \(y^2\) term: \(6y^2\)
- The \(y\) terms: \(13y + 11y = 24y\)
- The constant terms: \(-28 - 9 = -37\)
Combining these, we arrive at:
\[6y^2 + 24y - 37\]
This matches the first option provided in the problem.
Examples
In real-life scenarios, simplifying algebraic expressions is essential for efficient problem-solving in fields like finance, engineering, and computer programming. For example, if you are designing a budget where your total cost involves complex calculations like multiplying the number of items and their prices across different categories (represented by binomials), simplifying the expression allows you to quickly adjust your budget if the price or quantity of items changes. This saves time and minimizes the risk of calculation errors.