- The correct procedure involves expanding the squared binomial \((y-5)^2\) first, then distributing the \(-3\), and finally combining like terms to get \(-3y^2 + 37y - 84\).
- Statement 1 is false because exponents must be processed before distribution.
- Statement 2 is true because the resulting polynomial has three distinct terms.
- Statement 3 is false because the polynomial is degree 2, not 3.
- Statement 4 is false.
- Statement 5 is true.
- The true statements are: \(\boxed{\text{There are 3 terms in the simplified product, and the final simplified product is } -3y^2 + 37y - 84}\).
Explanation
- Evaluating the Expression and Initial Steps
To simplify the expression \(-3(y-5)^2 - 9 + 7y\), we must follow the order of operations. The first step is to evaluate the squared term \((y-5)^2\). Using the binomial expansion identity \((a-b)^2 = a^2 - 2ab + b^2\), we have \((y-5)^2 = y^2 - 2(y)(5) + 5^2 = y^2 - 10y + 25\). Thus, the expression becomes \(-3(y^2 - 10y + 25) - 9 + 7y\). Note that distributing the \(-3\) first, without squaring the binomial, would be incorrect because the exponent takes priority.
- Distributing and Combining Like Terms
Next, we distribute the \(-3\) to each term inside the parentheses: \(-3(y^2) - 3(-10y) - 3(25) = -3y^2 + 30y - 75\). Now, substitute this result back into the full expression: \((-3y^2 + 30y - 75) - 9 + 7y\). To simplify further, we group the like terms together: \(-3y^2 + (30y + 7y) + (-75 - 9)\). Calculating the sums, we obtain \(-3y^2 + 37y - 84\).
- Evaluating the Statements
Now we evaluate each of the given statements:
- "The first step in simplifying is to distribute the -3 throughout the parentheses." - This is False, as exponentiation precedes multiplication.
- "There are 3 terms in the simplified product." - This is True; the terms are \(-3y^2\), \(37y\), and \(-84\).
- "The simplified product is a degree 3 polynomial." - This is False; the highest power of \(y\) is \(2\), so it is a degree 2 polynomial.
- "The final simplified product is -3y^2 + 7y - 9." - This is False.
- "The final simplified product is -3y^2 + 37y - 84." - This is True.
Examples
In fields like engineering and physics, simplifying complex quadratic expressions is crucial for analyzing motion. For example, if you are modeling the height of a projectile over time using an expression like \(-3(t-5)^2 - 9 + 7t\), expanding and simplifying it allows you to easily identify the initial velocity and the starting height. Understanding the order of operations and the properties of polynomials ensures that the model remains accurate for predicting future positions of the object.