QuestionFactor the polynomial below. $64 a^{3}+27$

Studdy Solution

STEP 1

What is this asking? We need to rewrite this sum of cubes as a product of simpler expressions. Watch out! Don't forget to fully factor each term before applying the sum of cubes formula!

STEP 2

1. Recognize the Sum of Cubes
2. Apply the Formula
3. Simplify and Present the Result

STEP 3

Hey everyone!
Look closely at our expression: $64a^3 + 27$ .
Notice anything special about the numbers \textbf{64} and \textbf{27}?
They're perfect cubes!
We can rewrite \textbf{64} as $4^3$ and \textbf{27} as $3^3$ .
So, our expression becomes $(4a)^3 + 3^3$ .
This is a sum of cubes!

STEP 4

Remember the super helpful sum of cubes formula: $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$ .
In our case, $x$ is 4a and $y$ is 3.

STEP 5

Let's plug these values into the formula: $(4a)^3 + 3^3 = (4a + 3)((4a)^2 - (4a) \cdot 3 + 3^2)$

STEP 6

Now, let's simplify that second term. $(4a)^2$ becomes $16a^2$ , $(4a) \cdot 3$ is $12a$ , and $3^2$ is $9$ .

STEP 7

Putting it all together, we get: $(4a + 3)(16a^2 - 12a + 9)$ Look at that, perfectly factored!

STEP 8

The factored form of $64a^3 + 27$ is $(4a + 3)(16a^2 - 12a + 9)$ .

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