Evaluating the Algebraic Expression with Given Values

This problem involves evaluating a polynomial expression by substituting specific numerical values for the variables \(x\) and \(y\). We need to carefully perform the calculations, paying attention to exponents and signs.

The Polynomial Expression

The expression we need to evaluate is:

\[ 27x^3 + 58x^2y + 31xy^2 + 8y^3 \]

Given Variable Values

The values provided for the variables are:

• \(x = 5\)
• \(y = -7\)

Substituting the Values

We substitute \(x = 5\) and \(y = -7\) into the expression:

\[ 27(5)^3 + 58(5)^2(-7) + 31(5)(-7)^2 + 8(-7)^3 \]

Step-by-Step Calculation

Let's calculate each term individually:

Term 1: \(27x^3\)

• Substitute \(x=5\): \(27 \times (5)^3\)
• Calculate \(5^3\): \(5^3 = 5 \times 5 \times 5 = 125\)
• Multiply: \(27 \times 125 = 3375\)

Term 2: \(58x^2y\)

• Substitute \(x=5\) and \(y=-7\): \(58 \times (5)^2 \times (-7)\)
• Calculate \(5^2\): \(5^2 = 5 \times 5 = 25\)
• Multiply: \(58 \times 25 \times (-7) = 1450 \times (-7) = -10150\)

Term 3: \(31xy^2\)

• Substitute \(x=5\) and \(y=-7\): \(31 \times 5 \times (-7)^2\)
• Calculate \((-7)^2\): \((-7)^2 = (-7) \times (-7) = 49\)
• Multiply: \(31 \times 5 \times 49 = 155 \times 49 = 7595\)

Term 4: \(8y^3\)

• Substitute \(y=-7\): \(8 \times (-7)^3\)
• Calculate \((-7)^3\): \((-7)^3 = (-7) \times (-7) \times (-7) = 49 \times (-7) = -343\)
• Multiply: \(8 \times (-343) = -2744\)

Combining the Term Values

Now, we sum the results of the four terms:

\[ 3375 + (-10150) + 7595 + (-2744) \]

This simplifies to:

\[ 3375 - 10150 + 7595 - 2744 \]

Let's group the positive and negative terms:

\[ (3375 + 7595) - (10150 + 2744) \]

Calculate the sums:

\[ 10970 - 12894 \]

Perform the final subtraction:

\[ -1924 \]

Final Result

The calculated value of the expression \(27x^3 + 58x^2y + 31xy^2 + 8y^3\) when \(x = 5\) and \(y = -7\) is \(-1924\).