Factor using the formula for the sum or difference of two cubes. $x cubed minus 27$

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Recognize that the expression $x^3 - 27$ is a difference of cubes because $x^3$ is a cube and $27$ can be written as \$3^3$.

Recall the formula for the difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Identify $a = x$ and $b = 3$ in the expression $x^3 - 27$.

Apply the difference of cubes formula by substituting $a$ and $b$: $(x - 3)(x^2 + 3x + 9)$.

Write the fully factored form as the product of the binomial and trinomial: $(x - 3)(x^2 + 3x + 9)$.

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Difference of Cubes Formula

The difference of cubes formula is used to factor expressions of the form a³ - b³. It states that a³ - b³ = (a - b)(a² + ab + b²). Recognizing this pattern allows you to break down cubic expressions into simpler polynomial factors.

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To apply the sum or difference of cubes formula, you must identify terms that are perfect cubes. For example, x³ is a cube of x, and 27 is a cube of 3 since 3³ = 27. Recognizing these helps in correctly setting a and b in the formula.

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Factoring polynomials involves rewriting them as products of simpler polynomials. This process simplifies solving equations and analyzing functions. Using special formulas like the difference of cubes is a key technique in polynomial factoring.

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