Verified Answer :

  • Use the distributive property to multiply the constant -9 by each term in the trinomial: \(-9 \cdot (y^2 - 2y - 2)\).
  • Perform each multiplication: \(-9 \cdot y^2 = -9y^2\), \(-9 \cdot (-2y) = 18y\), and \(-9 \cdot (-2) = 18\).
  • Combine the terms to reach the final simplified expression: \(\boxed{-9y^2 + 18y + 18}\).

Explanation

  1. Analyze the expression and identify the method

The goal is to simplify the expression \(-9(y^2 - 2y - 2)\) by applying the distributive property. This property states that for any constants \(k, a, b,\) and \(c\), the expression \(k(a + b + c)\) is equivalent to \(ka + kb + kc\). In our case, \(k = -9\), \(a = y^2\), \(b = -2y\), and \(c = -2\).

  1. Apply the distributive property

Distribute the factor \(-9\) to each term inside the parentheses:
\[-9 \cdot (y^2) + (-9) \cdot (-2y) + (-9) \cdot (-2)\]
Now, compute each multiplication:

  1. \(-9 \cdot y^2 = -9y^2\)
  2. \(-9 \cdot (-2y) = 18y\) (since a negative times a negative is a positive)
  3. \(-9 \cdot (-2) = 18\) (since a negative times a negative is a positive)
  1. Formulate the final result

By combining the results of the multiplications from the previous step, we obtain:
\[-9y^2 + 18y + 18\]
This is the simplified form of the expression. The final answer is \(\boxed{-9y^2 + 18y + 18}\).

Examples

Distributive property is frequently used in finance when calculating interest or discounts over multiple items. For instance, if you purchase three separate items priced at \(\$x\), \(\$y\), and \(\$z\), and all of them are subject to a \(-10\%\) discount (represented as a multiplier of \(0.9\)), you can calculate the total price either by discounting each item individually, \((0.9 \times x) + (0.9 \times y) + (0.9 \times z)\), or by finding the sum first and applying the discount to the total, \(0.9 \times (x + y + z)\). Understanding this property helps simplify complex calculations and ensures accuracy when dealing with combined totals.

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