How To Factor X3 125?

How to Factor x^3 – 125

Have you ever wondered how to factor x^3 – 125? This is a common problem that students encounter in algebra, and it can be tricky to figure out. But don’t worry, we’re here to help! In this article, we’ll walk you through the steps of factoring x^3 – 125, and we’ll provide some tips and tricks to help you remember the process.

So if you’re ready to learn how to factor x^3 – 125, keep reading!

Step Explanation Example
1. Find two numbers that add up to 3 and multiply to 125. 5 and 25
2. Write the factors as x^3 + 5x^2 + 25x x^3 + 5x^2 + 25x
3. Factor out the common factor of x x(x^2 + 5x + 25)

In this tutorial, we will learn how to factor the polynomial $x^3 – 125$. We will first review the definition of the polynomial, its roots, and its factorization. Then, we will discuss three methods for factoring the polynomial: the grouping method, the difference-of-two-squares method, and the quadratic formula.

The Polynomial x3 – 125

The polynomial $x^3 – 125$ is a cubic polynomial. A cubic polynomial is a polynomial of degree 3, meaning that it has three terms. The terms of the polynomial are $x^3$, $-125$, and 0.

The roots of a polynomial are the values of $x$ that make the polynomial equal to 0. The roots of the polynomial $x^3 – 125$ are $5$, $-5$, and 0.

The factorization of a polynomial is the expression of the polynomial as a product of other polynomials. The factorization of the polynomial $x^3 – 125$ is $(x – 5)^2(x + 5)$.

Methods for Factoring x3 – 125

There are three methods for factoring the polynomial $x^3 – 125$: the grouping method, the difference-of-two-squares method, and the quadratic formula.

The grouping method is a method for factoring a polynomial by grouping terms together. To use the grouping method, we first group the terms of the polynomial into two groups:

$$
x^3 – 125 = (x^3 – 5x^2) + (5x^2 – 125)
$$

We then factor each group of terms using the distributive property:

$$
x^3 – 5x^2 = x^2(x – 5)
$$

$$
5x^2 – 125 = 5x^2 – 25x + 25x – 125
$$

$$
= 5x(x – 5) + 25(x – 5)
$$

We then combine the two factors to get the factorization of the polynomial:

$$
x^3 – 125 = (x – 5)^2(x + 5)
$$

The difference-of-two-squares method is a method for factoring a polynomial of the form $a^2 – b^2$. To use the difference-of-two-squares method, we first write the polynomial in the form $a^2 – b^2$:

$$
x^3 – 125 = (x^2)^2 – (5^2)^2
$$

We then factor the expression using the difference-of-two-squares formula:

$$
a^2 – b^2 = (a + b)(a – b)
$$

In this case, $a = x^2$ and $b = 5^2$. So, the factorization of the polynomial is:

$$
x^3 – 125 = (x^2 + 5^2)(x – 5^2)
$$

We can then simplify this to get the factorization of the polynomial:

$$
x^3 – 125 = (x + 5)(x – 5)^2
$$

The quadratic formula is a method for factoring a polynomial of the form $ax^2 + bx + c$. To use the quadratic formula, we first write the polynomial in the form $ax^2 + bx + c$:

$$
x^3 – 125 = x^3 – 125
$$

We then substitute the values of $a$, $b$, and $c$ into the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
$$

In this case, $a = 1$, $b = 0$, and $c = -125$. So, the quadratic formula gives us the following two roots:

$$
x = \frac{-0 \pm \sqrt{0^2 – 4(1)(-125)}}{2(1)}
$$

$$
= \frac{0 \pm \sqrt{500}}{2}
$$

$$
= \pm 5\sqrt{5}
$$

We can then write the factorization

How to Factor x3 – 125?

In this tutorial, we will show you how to factor the polynomial x3 – 125. This is a three-term polynomial, so we will use the grouping method to factor it.

Step 1: Find two numbers that add up to -1 and multiply to -125.

In this case, the two numbers are -5 and 25.

Step 2: Group the terms of the polynomial so that the first two terms have the common factor -5 and the last two terms have the common factor 25.

We can rewrite the polynomial as follows:

x3 – 125 = (x – 5)(x2 + 5x + 25)

Step 3: Factor the quadratic expression inside the parentheses.

We can use the quadratic formula to factor the quadratic expression inside the parentheses.

x2 + 5x + 25 = (x + 5)(x + 5)

Step 4: Simplify the expression.

We can simplify the expression by multiplying out the factors:

x3 – 125 = (x – 5)(x + 5)(x + 5)

Therefore, the factored form of x3 – 125 is (x – 5)(x + 5)(x + 5).

Example

Let’s use the factoring method we just learned to factor the polynomial x3 – 64.

Step 1: Find two numbers that add up to -1 and multiply to -64.

In this case, the two numbers are -8 and 8.

Step 2: Group the terms of the polynomial so that the first two terms have the common factor -8 and the last two terms have the common factor 8.

We can rewrite the polynomial as follows:

x3 – 64 = (x – 8)(x2 + 8x + 64)

Step 3: Factor the quadratic expression inside the parentheses.

We can use the quadratic formula to factor the quadratic expression inside the parentheses.

x2 + 8x + 64 = (x + 8)(x + 8)

Step 4: Simplify the expression.

We can simplify the expression by multiplying out the factors:

x3 – 64 = (x – 8)(x + 8)(x + 8)

Therefore, the factored form of x3 – 64 is (x – 8)(x + 8)(x + 8).

In this tutorial, we showed you how to factor the polynomial x3 – 125. We also showed you how to factor a general cubic polynomial. You can use the same method to factor any cubic polynomial.

Applications of Factoring x3 – 125

There are many applications of factoring x3 – 125. Here are a few examples:

  • Solving equations involving x3 – 125
  • Finding the zeros of a function
  • Simplifying expressions

Solving Equations Involving x3 – 125

One application of factoring x3 – 125 is solving equations involving this polynomial. For example, consider the equation x3 – 125 = 0. We can factor this equation as follows:

x3 – 125 = (x – 5)(x + 5)(x + 5) = 0

This equation has three solutions: -5, 5, and -5.

Finding the Zeros of a Function

Another application of factoring x3 – 125 is finding the zeros of a function. For example, consider the function f(x) = x3 – 125. We can factor this function as follows:

f(x) = x3 – 125 = (x – 5)(x + 5)(x + 5)

The zeros of this function are -5, 5, and -5.

Simplifying Expressions

A third application of factoring x3 – 125 is simplifying expressions. For example, consider the expression x3 – 125. We can factor this expression as follows:

x3 – 125 =

How do I factor x3 – 125?

1. First, we can factor out the greatest common factor (GCF) of 1 and x3 – 125, which is x. This gives us x(x2 – 125).
2. Next, we can use the difference of squares formula to factor x2 – 125 into (x + 15)(x – 15).
3. Therefore, the final factored form of x3 – 125 is x(x + 15)(x – 15).

What are the steps involved in factoring x3 – 125?

1. Step 1: Find the GCF of 1 and x3 – 125.
2. Step 2: Use the difference of squares formula to factor x2 – 125.
3. Step 3: Simplify the expression to get the final factored form.

Can you give me an example of factoring x3 – 125?

Sure, here is an example of factoring x3 – 125:

1. First, we find the GCF of 1 and x3 – 125, which is x. This gives us x(x2 – 125).
2. Next, we use the difference of squares formula to factor x2 – 125 into (x + 15)(x – 15).
3. Therefore, the final factored form of x3 – 125 is x(x + 15)(x – 15).

What are some tips for factoring x3 – 125?

Here are some tips for factoring x3 – 125:

  • Remember to find the GCF of 1 and x3 – 125 first. This will make the factoring process much easier.
  • Use the difference of squares formula to factor x2 – 125. This is a powerful tool that can be used to factor many different types of quadratic expressions.
  • Be careful not to make any mistakes when simplifying the expression. This can be easy to do, so it’s important to double-check your work.

What are some common mistakes people make when factoring x3 – 125?

Here are some common mistakes people make when factoring x3 – 125:

  • Forgetting to find the GCF of 1 and x3 – 125. This is a critical step in the factoring process, and it’s easy to forget to do it.
  • Not using the difference of squares formula to factor x2 – 125. This is a powerful tool that can be used to factor many different types of quadratic expressions.
  • Making mistakes when simplifying the expression. This can be easy to do, so it’s important to double-check your work.

    In this article, we have discussed how to factor x^3 – 125. We first used the sum of cubes identity to write x^3 – 125 as a difference of two cubes. Then, we used the difference of two cubes factoring formula to factor x^3 – 125 as (x – 5)(x^2 + 5x + 25). Finally, we verified our answer by multiplying the factors.

We hope that this article has been helpful. Please feel free to contact us if you have any questions.

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