Math / Algebra

QuestionFactor the following sum of two cubes. $9 x^{3}+72$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $9 x^{3}+72=$ $\square$ (Factor completely. Simplify your answer.) B. The polynomial is prime.

Studdy Solution

STEP 1

What is this asking? We're asked to factor the expression $9x^3 + 72$ , which means rewriting it as a product of simpler terms, if possible.
If it can't be factored, it's called a prime polynomial. Watch out! Don't forget to factor out any greatest common factor (GCF) before trying to apply any special factoring formulas!

STEP 2

1. Factor out the GCF
2. Apply the sum of cubes formula

STEP 3

Let's look at the expression $9x^3 + 72$ .
We need to find the greatest common factor (GCF) of the two terms.
The GCF of $9$ and $72$ is 9, so we can factor out a $9$ from both terms.

STEP 4

Factoring out the $9$ gives us: $9x^3 + 72 = 9(x^3 + 8)$ Remember, factoring out a $9$ is the same as dividing each term by $9$ .
We can check our work by distributing the $9$ back: $9 \cdot x^3 + 9 \cdot 8 = 9x^3 + 72$ .
Perfect!

STEP 5

Now, we have $9(x^3 + 8)$ .
Notice that $8$ can be written as $2^3$ .
So, we can rewrite the expression inside the parentheses as a sum of cubes: $x^3 + 8 = x^3 + 2^3$ This is exciting!
We can use the sum of cubes formula, which says: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

STEP 6

In our case, $a = x$ and $b = 2$ .
Let's apply the formula: $x^3 + 2^3 = (x+2)(x^2 - x \cdot 2 + 2^2)$ $x^3 + 2^3 = (x+2)(x^2 - 2x + 4)$ Don't forget about the $9$ we factored out earlier!

STEP 7

Putting it all together, our fully factored expression is: $9(x+2)(x^2 - 2x + 4)$

STEP 8

The fully factored form of $9x^3 + 72$ is $9(x+2)(x^2 - 2x + 4)$ .
So the answer is A. $9x^3 + 72 = 9(x+2)(x^2 - 2x + 4)$ .

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QuestionFactor the following sum of two cubes. $9 x^{3}+72$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $9 x^{3}+72=$ $\square$ (Factor completely. Simplify your answer.) B. The polynomial is prime.

Studdy Solution

STEP 1

STEP 2

1. Factor out the GCF
2. Apply the sum of cubes formula

STEP 3

Let's look at the expression $9x^3 + 72$ .
We need to find the greatest common factor (GCF) of the two terms.
The GCF of $9$ and $72$ is 9, so we can factor out a $9$ from both terms.

STEP 4

Factoring out the $9$ gives us: $9x^3 + 72 = 9(x^3 + 8)$ Remember, factoring out a $9$ is the same as dividing each term by $9$ .
We can check our work by distributing the $9$ back: $9 \cdot x^3 + 9 \cdot 8 = 9x^3 + 72$ .
Perfect!

STEP 5

Now, we have $9(x^3 + 8)$ .
Notice that $8$ can be written as $2^3$ .
So, we can rewrite the expression inside the parentheses as a sum of cubes: $x^3 + 8 = x^3 + 2^3$ This is exciting!
We can use the sum of cubes formula, which says: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

STEP 6

In our case, $a = x$ and $b = 2$ .
Let's apply the formula: $x^3 + 2^3 = (x+2)(x^2 - x \cdot 2 + 2^2)$ $x^3 + 2^3 = (x+2)(x^2 - 2x + 4)$ Don't forget about the $9$ we factored out earlier!

STEP 7

Putting it all together, our fully factored expression is: $9(x+2)(x^2 - 2x + 4)$

STEP 8

The fully factored form of $9x^3 + 72$ is $9(x+2)(x^2 - 2x + 4)$ .
So the answer is A. $9x^3 + 72 = 9(x+2)(x^2 - 2x + 4)$ .

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QuestionFactor the following sum of two cubes. 9x3+729 x^{3}+729x3+72Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 9x3+72=9 x^{3}+72=9x3+72= □\square□ (Factor completely. Simplify your answer.) B. The polynomial is prime.

Studdy Solution

STEP 1

STEP 2

1. Factor out the GCF2. Apply the sum of cubes formula

STEP 3

Let's **look** at the expression 9x3+729x^3 + 729x3+72.We need to find the **greatest common factor** (GCF) of the two terms.The GCF of 999 and 727272 is **9**, so we can **factor out** a 999 from both terms.

STEP 4

**Factoring out** the 999 gives us: 9x3+72=9(x3+8)9x^3 + 72 = 9(x^3 + 8)9x3+72=9(x3+8) Remember, factoring out a 999 is the same as dividing each term by 999.We can check our work by distributing the 999 back: 9⋅x3+9⋅8=9x3+729 \cdot x^3 + 9 \cdot 8 = 9x^3 + 729⋅x3+9⋅8=9x3+72.Perfect!

STEP 5

STEP 6

In our case, a=xa = xa=x and b=2b = 2b=2.Let's **apply** the formula: x3+23=(x+2)(x2−x⋅2+22)x^3 + 2^3 = (x+2)(x^2 - x \cdot 2 + 2^2)x3+23=(x+2)(x2−x⋅2+22) x3+23=(x+2)(x2−2x+4)x^3 + 2^3 = (x+2)(x^2 - 2x + 4)x3+23=(x+2)(x2−2x+4)Don't forget about the 999 we factored out earlier!

STEP 7

Putting it all together, our **fully factored expression** is: 9(x+2)(x2−2x+4)9(x+2)(x^2 - 2x + 4)9(x+2)(x2−2x+4)

STEP 8

The fully factored form of 9x3+729x^3 + 729x3+72 is 9(x+2)(x2−2x+4)9(x+2)(x^2 - 2x + 4)9(x+2)(x2−2x+4).So the answer is A. 9x3+72=9(x+2)(x2−2x+4)9x^3 + 72 = 9(x+2)(x^2 - 2x + 4)9x3+72=9(x+2)(x2−2x+4).

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFactor the following sum of two cubes. $9 x^{3}+72$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $9 x^{3}+72=$ $\square$ (Factor completely. Simplify your answer.) B. The polynomial is prime.

1. Factor out the GCF
2. Apply the sum of cubes formula

Let's look at the expression $9x^3 + 72$ .
We need to find the greatest common factor (GCF) of the two terms.
The GCF of $9$ and $72$ is 9, so we can factor out a $9$ from both terms.

Factoring out the $9$ gives us: $9x^3 + 72 = 9(x^3 + 8)$ Remember, factoring out a $9$ is the same as dividing each term by $9$ .
We can check our work by distributing the $9$ back: $9 \cdot x^3 + 9 \cdot 8 = 9x^3 + 72$ .
Perfect!

In our case, $a = x$ and $b = 2$ .
Let's apply the formula: $x^3 + 2^3 = (x+2)(x^2 - x \cdot 2 + 2^2)$ $x^3 + 2^3 = (x+2)(x^2 - 2x + 4)$ Don't forget about the $9$ we factored out earlier!

Putting it all together, our fully factored expression is: $9(x+2)(x^2 - 2x + 4)$

The fully factored form of $9x^3 + 72$ is $9(x+2)(x^2 - 2x + 4)$ .
So the answer is A. $9x^3 + 72 = 9(x+2)(x^2 - 2x + 4)$ .