Verified Answer :

  • Apply the distributive property to multiply each term of the trinomial \((2y^2 + \frac{3}{4}y - 7)\) by the terms of the binomial \((\frac{1}{2}y + 3)\).
  • Calculate the individual products: \(y^3 + 6y^2 + \frac{3}{8}y^2 + \frac{9}{4}y - \frac{7}{2}y - 21\).
  • Group like terms and simplify: \(y^3 + (6 + \frac{3}{8})y^2 + (\frac{9}{4} - \frac{14}{4})y - 21\).
  • The final expanded expression is \(\boxed{y^3 + \frac{51}{8}y^2 - \frac{5}{4}y - 21}\).

Explanation

  1. Problem Analysis

The problem requires finding the product of a quadratic trinomial \((2y^2 + \frac{3}{4}y - 7)\) and a linear binomial \((\frac{1}{2}y + 3)\). To achieve this, we use the distributive property, ensuring that every term of the first polynomial is multiplied by every term of the second polynomial. This approach guarantees that no terms are omitted during the expansion process.

  1. Applying the Distributive Property

We distribute the terms as follows:
\[(2y^2 + \frac{3}{4}y - 7)(\frac{1}{2}y + 3) = 2y^2(\frac{1}{2}y + 3) + \frac{3}{4}y(\frac{1}{2}y + 3) - 7(\frac{1}{2}y + 3)\]
Performing the multiplication for each part:

  1. \(2y^2 \cdot \frac{1}{2}y = y^3\)
  2. \(2y^2 \cdot 3 = 6y^2\)
  3. \(\frac{3}{4}y \cdot \frac{1}{2}y = \frac{3}{8}y^2\)
  4. \(\frac{3}{4}y \cdot 3 = \frac{9}{4}y\)
  5. \(-7 \cdot \frac{1}{2}y = -\frac{7}{2}y\)
  6. \(-7 \cdot 3 = -21\)
  1. Grouping and Combining Like Terms

Now, we collect all the terms and group them by their degree of \(y\):
\[y^3 + (6y^2 + \frac{3}{8}y^2) + (\frac{9}{4}y - \frac{7}{2}y) - 21\]
Next, simplify the coefficients for \(y^2\) and \(y\):

  • For \(y^2\): \(6 + \frac{3}{8} = \frac{48}{8} + \frac{3}{8} = \frac{51}{8}\)
  • For \(y\): \(\frac{9}{4} - \frac{7}{2} = \frac{9}{4} - \frac{14}{4} = -\frac{5}{4}\)

Thus, the expanded expression is:
\[y^3 + \frac{51}{8}y^2 - \frac{5}{4}y - 21\]

Examples

Expanding polynomials is a fundamental skill in algebra with many practical applications. For instance, in engineering and physics, when calculating the volume of a geometric object where dimensions are expressed as linear expressions of a variable (e.g., width, height, and length), multiplying these expressions together allows for the determination of the volume as a single polynomial function. This function can then be used to analyze how the volume changes relative to the variable, aiding in design optimization and resource planning.

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