Factorizing the Expression $p^3 + 27$

The question asks us to find the factors of the algebraic expression $p^3 + 27$. This expression is a classic example of the sum of two cubes.

Understanding the Sum of Cubes Formula

To factorize the expression $p^3 + 27$, we can use the algebraic identity for the sum of cubes. The formula states:

$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$

This formula allows us to break down the sum of two cubed terms into a product of two factors: a linear factor $(a+b)$ and a quadratic factor $(a^2 - ab + b^2)$.

Applying the Formula to $p^3 + 27$

Let's identify the terms $a$ and $b$ in our expression $p^3 + 27$. We can rewrite the expression as:

$$ p^3 + 27 = p^3 + 3^3 $$

Comparing this with the general formula $a^3 + b^3$, we can see that:

• $a^3 = p^3$, which implies $a = p$.
• $b^3 = 3^3$, which implies $b = 3$.

Now, we substitute these values of $a$ and $b$ into the sum of cubes formula:

$$ p^3 + 3^3 = (p + 3)(p^2 - (p)(3) + 3^2) $$

Simplifying the terms in the second factor:

• $p \times 3 = 3p$
• $3^2 = 9$

Substituting these simplified terms back into the factored expression, we get:

$$ p^3 + 27 = (p + 3)(p^2 - 3p + 9) $$

Comparing the Result with Options

Let's examine the given options to find the one that matches our result:

• Option 1: $(p - 3) (p^2 - 3p + 9)$ - This is incorrect because the first factor should be $(p + 3)$.
• Option 2: $(p + 3) (p^2 + 3p + 9)$ - This is incorrect. The middle term in the second factor should be $-3p$, not $+3p$.
• Option 3: $(p + 3) (p^2 - 3p + 9)$ - This matches our calculated factorization perfectly.
• Option 4: $(p + 3) (p^2 - 3p - 9)$ - This is incorrect because the constant term in the second factor should be $+9$, not $-9$.

Therefore, the correct factorization of the expression $p^3 + 27$ is $(p + 3)(p^2 - 3p + 9)$.