Verified Answer :

  • Identify the polynomials to be added: \(P_1(g) = 5g^2 + 7g\) and \(P_2(g) = 3g^3 + g^2 + 2g\).
  • Group the terms with the same exponents: cubic, quadratic, and linear.
  • Combine the coefficients for each degree: \(3g^3\), \((5+1)g^2\), and \((7+2)g\).
  • State the final sum: \(\boxed{3g^3 + 6g^2 + 9g}\).

Explanation

  1. Analysis of the Polynomials

We are tasked with adding two polynomials, \(P_1(g) = 5g^2 + 7g\) and \(P_2(g) = 3g^3 + g^2 + 2g\). To find the sum \(S(g) = P_1(g) + P_2(g)\), we identify and group the like terms, which are terms that have the same variable raised to the same power. In this expression, the terms are \(3g^3\), \(5g^2\), \(g^2\), \(7g\), and \(2g\).

  1. Grouping Like Terms

Next, we group the like terms together based on the degree of the variable \(g\):

  • The cubic term (degree 3): \(3g^3\)
  • The quadratic terms (degree 2): \(5g^2\) and \(1g^2\)
  • The linear terms (degree 1): \(7g\) and \(2g\)

Writing the sum expression, we have:
\[ S(g) = 3g^3 + (5g^2 + g^2) + (7g + 2g) \]

  1. Combining Coefficients and Final Result

Now, we simplify by adding the coefficients of the grouped like terms:

  • For the quadratic terms: \(5g^2 + 1g^2 = (5+1)g^2 = 6g^2\)
  • For the linear terms: \(7g + 2g = (7+2)g = 9g\)

Combining these results, the final simplified sum is:
\[ S(g) = 3g^3 + 6g^2 + 9g \]

Examples

Polynomial addition is widely used in physics and economics. For example, if the cost of manufacturing a product is defined by two different components—say, fixed material costs \(M(g) = 5g^2 + 7g\) and variable labor costs \(L(g) = 3g^3 + g^2 + 2g\), where \(g\) represents the number of units produced—the total cost function \(T(g) = M(g) + L(g)\) is calculated by adding these polynomials. Mastering this helps businesses calculate overhead, project expenses, and understand how production scale affects their bottom line.

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