Determine If The Following Expression Is A Sum Or Difference Of Cubes:${ (x-6)\left(x^2+12x+36\right) }$Is The Answer A Sum/difference Of Cubes? A. YES B. NO

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Introduction

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In algebra, a sum or difference of cubes is a mathematical expression that can be factored into the product of three binomials. This concept is crucial in solving polynomial equations and is a fundamental aspect of algebraic manipulation. In this article, we will explore whether the given expression $(x-6)\left(x^2+12x+36\right)$ is a sum or difference of cubes.

What is a Sum or Difference of Cubes?

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A sum of cubes is a mathematical expression of the form $a^3 + b^3$, which can be factored into the product of three binomials as $(a+b)(a^2-ab+b^2)$. On the other hand, a difference of cubes is an expression of the form $a^3 - b^3$, which can be factored into the product of three binomials as $(a-b)(a^2+ab+b^2)$.

Factoring the Given Expression

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To determine whether the given expression is a sum or difference of cubes, we need to factor it. Let's start by factoring the quadratic expression $x^2+12x+36$. We can use the perfect square trinomial formula to factor it:

$$x^2+12x+36 = (x+6)^2$$

Now, we can rewrite the given expression as:

$$(x-6)\left(x^2+12x+36\right) = (x-6)(x+6)^2$$

Is the Expression a Sum or Difference of Cubes?

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To determine whether the expression is a sum or difference of cubes, we need to check if it can be factored into the product of three binomials. Let's examine the expression $(x-6)(x+6)^2$:

$$(x-6)(x+6)^2 = (x-6)(x^2+12x+36)$$

We can see that the expression is not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in the form of a sum or difference of cubes. However, we can rewrite it as:

$$(x-6)(x+6)^2 = (x-6)(x+6)(x+6) = (x-6)(x+6)(x+6)$$

This expression is still not in

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Q: What is a sum or difference of cubes?

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A: A sum of cubes is a mathematical expression of the form $a^3 + b^3$, which can be factored into the product of three binomials as $(a+b)(a^2-ab+b^2)$. On the other hand, a difference of cubes is an expression of the form $a^3 - b^3$, which can be factored into the product of three binomials as $(a-b)(a^2+ab+b^2)$.

Q: How do I determine if an expression is a sum or difference of cubes?

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A: To determine if an expression is a sum or difference of cubes, you need to check if it can be factored into the product of three binomials. You can use the following steps:

1. Check if the expression is in the form of $a^3 + b^3$ or $a^3 - b^3$.
2. If it is, try to factor it using the formulas for sum and difference of cubes.
3. If it can be factored, then it is a sum or difference of cubes.

Q: What are the formulas for sum and difference of cubes?

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A: The formulas for sum and difference of cubes are:

• Sum of cubes: $a^3 + b^3 = (a+b)(a^2-ab+b^2)$
• Difference of cubes: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$

Q: How do I factor a sum or difference of cubes?

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A: To factor a sum or difference of cubes, you need to use the formulas for sum and difference of cubes. Here are the steps:

1. Identify the values of $a$ and $b$ in the expression.
2. Plug these values into the formulas for sum and difference of cubes.
3. Simplify the resulting expression to get the factored form.

Q: What are some examples of sum and difference of cubes?

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A: Here are some examples of sum and difference of cubes:

• Sum of cubes: $x^3 + 8^3 = (x+8)(x^2-8x+64)$
• Difference of cubes: $x^3 - 27^3 = (x-27)(x^2+27x+729)$

Q: Can I use the formulas for sum and difference of cubes to factor any expression?

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A: No, the formulas for sum and difference of cubes can only be used to factor expressions that are in the form of $a^3 + b^3$ or $a^3 - b^3$. If an expression is not in this form, you will need to use other factoring techniques.

Q: What are some common mistakes to avoid when factoring sum and difference of cubes?

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A: Here are some common mistakes to avoid when factoring sum and difference of cubes:

• Not identifying the values of $a$ and $b$ correctly.
• Not using the correct formulas for sum and difference of cubes.
• Not simplifying the resulting expression correctly.

Q: How can I practice factoring sum and difference of cubes?

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A: You can practice factoring sum and difference of cubes by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some real-world applications of sum and difference of cubes?

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A: Sum and difference of cubes have many real-world applications, including:

• Algebraic manipulation: Sum and difference of cubes are used to simplify and manipulate algebraic expressions.
• Calculus: Sum and difference of cubes are used to solve optimization problems and find the maximum or minimum value of a function.
• Physics: Sum and difference of cubes are used to describe the motion of objects and the behavior of physical systems.

Q: Can I use sum and difference of cubes to factor any polynomial expression?

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A: No, sum and difference of cubes can only be used to factor polynomial expressions that are in the form of $a^3 + b^3$ or $a^3 - b^3$. If a polynomial expression is not in this form, you will need to use other factoring techniques.