What Is The Factored Form Of This Expression? 27 M 3 + 125 N 3 27m^3 + 125n^3 27 M 3 + 125 N 3 Drag The Factors To The Correct Locations On The Image. Not All Factors Will Be Used.Options:1. $9m^2 + 15mn + 25n^2$2. $9m^2 - 15mn + 25n^2$3. $3m^2 - 8mn +

Understanding the Problem

The given expression is $27m^3 + 125n^3$. We are asked to find the factored form of this expression and drag the factors to the correct locations on the image. This problem involves factoring a sum of cubes, which is a common algebraic expression.

Factoring a Sum of Cubes

A sum of cubes is an expression of the form $a^3 + b^3$. To factor this expression, we can use the formula:

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

This formula allows us to factor the sum of cubes into two binomials.

Applying the Formula

In the given expression $27m^3 + 125n^3$, we can see that both terms are perfect cubes. We can rewrite the expression as:

$$(3m)^3 + (5n)^3$$

Now, we can apply the formula to factor the expression:

$$(3m)^3 + (5n)^3 = (3m + 5n)((3m)^2 - (3m)(5n) + (5n)^2)$$

Simplifying the Expression

We can simplify the expression further by expanding the squared term:

$$(3m)^2 - (3m)(5n) + (5n)^2 = 9m^2 - 15mn + 25n^2$$

Conclusion

Therefore, the factored form of the expression $27m^3 + 125n^3$ is:

$$(3m + 5n)(9m^2 - 15mn + 25n^2)$$

Dragging the Factors to the Correct Locations

Based on the factored form, we can drag the factors to the correct locations on the image. The correct locations are:

• Factor 1: $9m^2 - 15mn + 25n^2$
• Factor 2: $3m + 5n$

Discussion

This problem involves factoring a sum of cubes, which is a common algebraic expression. The formula for factoring a sum of cubes is:

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

This formula allows us to factor the sum of cubes into two binomials. In the given expression $27m^3 + 125n^3$, we can see that both terms are perfect cubes. We can rewrite the expression as:

$$(3m)^3 + (5n)^3$$

Now, we can apply the formula to factor the expression:

$$(3m)^3 + (5n)^3 = (3m + 5n)((3m)^2 - (3m)(5n) + (5n)^2)$$

We can simplify the expression further by expanding the squared term:

$$(3m)^2 - (3m)(5n) + (5n)^2 = 9m^2 - 15mn + 25n^2$$

Therefore, the factored form of the expression $27m^3 + 125n^3$ is:

$$(3m + 5n)(9m^2 - 15mn + 25n^2)$$

Options

Based on the factored form, we can see that the correct options are:

• Option 1: $9m^2 - 15mn + 25n^2$
• Option 2: $3m + 5n$

Conclusion

In conclusion, the factored form of the expression $27m^3 + 125n^3$ is:

$$(3m + 5n)(9m^2 - 15mn + 25n^2)$$

This problem involves factoring a sum of cubes, which is a common algebraic expression. The formula for factoring a sum of cubes is:

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

This formula allows us to factor the sum of cubes into two binomials.

Understanding the Problem

In the previous article, we discussed how to factor a sum of cubes using the formula:

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

This formula allows us to factor the sum of cubes into two binomials. In this article, we will answer some common questions related to factoring a sum of cubes.

Q: What is a sum of cubes?

A: A sum of cubes is an expression of the form $a^3 + b^3$. It is a common algebraic expression that can be factored using the formula:

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

Q: How do I know if an expression is a sum of cubes?

A: To determine if an expression is a sum of cubes, look for the presence of perfect cubes. If both terms in the expression are perfect cubes, then it is a sum of cubes.

Q: What is the formula for factoring a sum of cubes?

A: The formula for factoring a sum of cubes is:

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

Q: How do I apply the formula to factor a sum of cubes?

A: To apply the formula, identify the values of $a$ and $b$ in the expression. Then, substitute these values into the formula:

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

Q: What if the expression is not a perfect cube?

A: If the expression is not a perfect cube, then it cannot be factored using the formula for sum of cubes. You may need to use other factoring techniques, such as factoring by grouping or factoring out a greatest common factor.

Q: Can I factor a difference of cubes using the same formula?

A: No, the formula for factoring a sum of cubes does not work for a difference of cubes. A difference of cubes is an expression of the form $a^3 - b^3$. To factor a difference of cubes, you can use the formula:

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

Q: What if I have a sum of cubes with variables?

A: If you have a sum of cubes with variables, you can still use the formula to factor it. For example, if you have the expression $x^3 + 8$, you can factor it as:

$$(x + 2)(x^2 - 2x + 4)$$

Q: Can I factor a sum of cubes with negative numbers?

A: Yes, you can factor a sum of cubes with negative numbers. For example, if you have the expression $-27x^3 + 125$, you can factor it as:

$$(-3x + 5)(9x^2 + 15x + 25)$$

Conclusion

In conclusion, factoring a sum of cubes is a common algebraic expression that can be factored using the formula:

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

This formula allows us to factor the sum of cubes into two binomials. By understanding the formula and how to apply it, you can easily factor sums of cubes and solve algebraic expressions.

Common Mistakes to Avoid

• Not recognizing a sum of cubes
• Not applying the formula correctly
• Not simplifying the expression after factoring
• Not checking for errors in the factored form

Tips and Tricks

• Always look for perfect cubes when factoring a sum of cubes
• Use the formula to factor a sum of cubes
• Simplify the expression after factoring
• Check for errors in the factored form

Practice Problems

• Factor the sum of cubes: $64 + 27x^3$
• Factor the sum of cubes: $125 + 8x^3$
• Factor the sum of cubes: $27x^3 + 64$

Solutions

• $64 + 27x^3 = (4 + 3x)(16 - 12x + 9x^2)$
• $125 + 8x^3 = (5 + 2x)(25 - 10x + 4x^2)$
• $27x^3 + 64 = (3x + 4)(9x^2 - 12x + 16)$