Questions: Factor the binomial: y^3-27

Factor the binomial: y^3-27
Transcript text: Factor the binomial: $y^{3}-27$
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Solution

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Solution Steps

Step 1: Identify the Polynomial

We are given the binomial \( y^{3} - 27 \). This expression can be recognized as a difference of cubes, since \( 27 \) can be expressed as \( 3^3 \).

Step 2: Apply the Difference of Cubes Formula

The difference of cubes can be factored using the formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] In our case, let \( a = y \) and \( b = 3 \). Thus, we can rewrite the expression as: \[ y^{3} - 3^{3} \]

Step 3: Factor the Expression

Applying the difference of cubes formula, we have: \[ y^{3} - 27 = (y - 3)(y^{2} + 3y + 9) \]

Final Answer

The factorized form of the polynomial \( y^{3} - 27 \) is: \[ \boxed{(y - 3)(y^{2} + 3y + 9)} \]

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