Verified Answer :

  • Multiply the binomials \((4y + 7)\) and \((y - 2)\) to get \(4y^2 - y - 14\).
  • Distribute \(6y\) across the resulting trinomial.
  • Combine the terms to reach the final simplified polynomial.
  • The result of the multiplication is \(\boxed{24y^3 - 6y^2 - 84y}\).

Explanation

  1. Problem Analysis

We are tasked with multiplying and simplifying the expression \(6y(4y + 7)(y - 2)\). First, we will expand the product of the two binomials \((4y + 7)\) and \((y - 2)\) using the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last). Then, we will multiply the resulting trinomial by the factor \(6y\). The second expression, \(24y^3 + 6y^2 - 84\), is already in its expanded polynomial form and does not require further simplification.

  1. Multiplying the Binomials

First, we multiply the two binomials:
\((4y + 7)(y - 2) = 4y(y) + 4y(-2) + 7(y) + 7(-2)\)
Calculating each term:
\(= 4y^2 - 8y + 7y - 14\)
Combining like terms (\(-8y\) and \(7y\)):
\(= 4y^2 - y - 14\)

  1. Multiplying by the Monomial Factor

Now, we multiply the entire resulting trinomial by \(6y\):
\(6y(4y^2 - y - 14) = 6y(4y^2) + 6y(-y) + 6y(-14)\)
Calculating each term:
\(= 24y^3 - 6y^2 - 84y\)
The final simplified expression is \(24y^3 - 6y^2 - 84y\).

Examples

Polynomial expansion is essential in various fields, such as engineering and physics, where complex systems are modeled using algebraic expressions. For instance, if you are calculating the volume of a storage container whose dimensions are defined as linear functions of a variable \(y\), multiplying those dimensions will result in a polynomial expression representing the volume as a function of \(y\). Knowing how to expand and simplify these expressions allows you to find the total volume efficiently, which is crucial for logistics and spatial planning.

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