Question

find the indicated sum.

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$$\begin{array}{r} 6x^3 + 6x^2 \\phantom{ - 5x} + 21 \\\\ 3x^3 - 2x^2 - 5x - 9 \\\\ \\hline \\end{array}$$

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Explanation:

Identify the polynomials to add

We are given two polynomials arranged vertically to find their sum:

$$
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This represents the operation of Polynomial Addition.

Align and group like terms

To add these polynomials, we group the terms with the same variable and exponent:

• Cubic terms (\(x^3\)): \(6x^3\) and \(3x^3\)
• Quadratic terms (\(x^2\)): \(6x^2\) and \(-2x^2\)
• Linear terms (\(x\)): \(0x\) and \(-5x\)
• Constant terms: \(21\) and \(-9\)

This systematic grouping is key to Polynomial Simplification.

Combine the coefficients

Now we add the coefficients for each group of like terms:

• For \(x^3\): \((6 + 3)x^3 = 9x^3\)
• For \(x^2\): \((6 - 2)x^2 = 4x^2\)
• For \(x\): \((0 - 5)x = -5x\)
• For constants: \(21 - 9 = 12\)

Write the final simplified polynomial

Combining all the simplified parts gives the final sum:

$$ 9x^3 + 4x^2 - 5x + 12 $$

Answer:

Find the indicated sum.

$$
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$$

The sum is <blank>\(9x^3 + 4x^2 - 5x + 12\)</blank>.