What Is The Factored Form Of 3X + 24Y?

The factored form of the equation 8x² + 12x is 4x(2x + 3).

Factoring a polynomial means expressing it as a product of simpler polynomials. To factor 8x² + 12x, we can first find the greatest common factor (GCF) of the two terms. The GCF of 8x² and 12x is 4x. This means we can rewrite the expression as 4x(2x + 3). We can check our answer by distributing the 4x back into the parentheses: 4x(2x + 3) = 8x² + 12x. This shows that our factored form is correct.

Let’s break down the process of finding the factored form of 8x² + 12x in more detail:

1. Identify the greatest common factor (GCF): The GCF of 8x² and 12x is 4x. This is because 4 is the greatest common factor of 8 and 12, and x is the common variable in both terms.
2. Factor out the GCF: Divide each term in the expression by the GCF, 4x:
8x² ÷ 4x = 2x
12x ÷ 4x = 3
3. Write the factored expression: The factored expression is 4x(2x + 3).

Remember that factoring is about finding the building blocks of a polynomial expression. This process helps us simplify expressions and understand their underlying structure.

The completely factored form of the expression 16x² + 8x + 32 is 8(2x² + x + 4).

Let’s break down how we get to this factored form.

First, we look for the greatest common factor (GCF) of the three terms in the expression. The GCF of 16, 8, and 32 is 8. We can factor out an 8 from each term:

16x² = 8 * 2x²
8x = 8 * x
32 = 8 * 4

Now we can rewrite the original expression as:

8 * 2x² + 8 * x + 8 * 4

Since we have an 8 in each term, we can factor it out:

8(2x² + x + 4)

This is the completely factored form of the expression 16x² + 8x + 32. We cannot factor the trinomial 2x² + x + 4 further using real numbers because there are no two numbers that multiply to 8 (2 times 4) and add up to 1 (the coefficient of the x term).

Let’s break down factoring the expression 3x + 24y.

The factored form of 3x + 24y is 3(x + 8y). We achieve this by finding the greatest common factor (GCF) of 3x and 24y, which is 3.

Here’s how the factoring process works:

1. Identify the GCF: The greatest common factor of 3x and 24y is 3.
2. Factor out the GCF: Divide each term in the expression by the GCF:
3x / 3 = x
24y / 3 = 8y
3. Write the factored expression: Combine the GCF and the results of the division within parentheses: 3(x + 8y)

Understanding the Concept of Factoring:

Factoring is a fundamental concept in algebra, particularly when dealing with polynomials. Essentially, it involves breaking down a complex expression into simpler components. The goal of factoring is to express a given expression as a product of its factors.

In the case of 3x + 24y, we’ve factored it into 3(x + 8y). This means we’ve expressed the original expression as the product of two factors: 3 and (x + 8y). This factorization can be useful for various algebraic operations, such as simplifying expressions, solving equations, and analyzing the behavior of functions.

By understanding the principles of factoring, you gain a deeper understanding of how algebraic expressions work and how they can be manipulated to solve problems.

Let’s break down how to write a polynomial in factored form. It’s basically like taking a polynomial and splitting it into a multiplication problem. You’re essentially finding the building blocks of the polynomial.

Think of it like this: You have a cake (the polynomial), and you want to figure out what ingredients (the factors) went into making it.

To write a polynomial in factored form, you need to express it as a product of simpler terms. These terms can be constant, linear (like *x + 2*), or even other polynomials that can’t be simplified further.

Here’s a simple example:

The polynomial x² + 5x + 6 can be factored as (x + 2)(x + 3).

Here’s how it works:

(x + 2) and (x + 3) are the factors of the polynomial.
* When you multiply these factors together, you get the original polynomial x² + 5x + 6.

Why is factored form useful?

Finding roots: The factored form helps you find the roots (or solutions) of the polynomial equation. The roots are the values of *x* that make the polynomial equal to zero. For example, in the factored form (x + 2)(x + 3), the roots are *x = -2* and *x = -3*.
Simplifying expressions: Factored form can make it easier to simplify complex polynomial expressions. For example, if you have a polynomial with a common factor, you can factor it out to simplify the expression.

Let’s explore how to factor different types of polynomials:

Factoring quadratics: A quadratic polynomial is a polynomial of degree 2, meaning its highest exponent is 2. You can use various techniques to factor quadratics, such as:
Factoring by grouping: Look for common factors within the polynomial and group them together.
The “AC” method: Multiply the coefficient of the *x²* term (the “A” term) by the constant term (the “C” term), then find two numbers that add up to the coefficient of the *x* term (the “B” term) and multiply to the “AC” product.
Factoring polynomials with more than two terms: For polynomials with more than two terms, you can try to find common factors, use the grouping method, or use more advanced factoring techniques like the “rational root theorem” or “synthetic division.”

Remember, practice makes perfect! The more you practice factoring polynomials, the more confident you’ll become in applying these techniques.

Let’s break down how to find the completely factored form of x⁴ + 8x² – 9.

The completely factored form of x⁴ + 8x² – 9 is (x + 3i)(x – 3i)(x + 1)(x – 1).

Here’s how we get there:

1. Recognize the pattern: Notice that this expression is a quadratic in disguise. If we let y = x², we can rewrite it as y² + 8y – 9.

2. Factor the quadratic: This quadratic factors nicely: (y + 9)(y – 1).

3. Substitute back: Now we substitute x² back in for y: (x² + 9)(x² – 1).

4. Factor further: The second factor, x² – 1, is a difference of squares and factors as (x + 1)(x – 1). The first factor, x² + 9, is also a difference of squares, but with imaginary numbers: (x + 3i)(x – 3i).

Therefore, the completely factored form of x⁴ + 8x² – 9 is (x + 3i)(x – 3i)(x + 1)(x – 1).

Understanding Imaginary Numbers

Imaginary numbers, like 3i, are essential for factoring expressions like x² + 9. They allow us to express the square roots of negative numbers. Remember that the square root of -1 is denoted by the imaginary unit i. So, the square root of 9 is 3i.

Why is this important?

When we factor x² + 9, we’re essentially asking “what numbers, when squared, add up to 9?” Since no real number squared equals -9, we need to introduce imaginary numbers. This concept helps us express the complete factorization of expressions that might not factor neatly using only real numbers.

In summary, factoring expressions with imaginary numbers allows us to express a complete factorization, revealing the full structure of the polynomial.

The fully factored form of the algebraic expression 32a³ + 12a² is 4a²(8a + 3).

Let’s break down how we arrive at this answer. Factoring an expression means finding the factors that, when multiplied together, give us the original expression. In this case, we need to find the greatest common factor (GCF) of 32a³ and 12a².

The GCF is the largest factor that divides into both terms. Let’s look at each term separately:

32a³: The factors of 32 are 1, 2, 4, 8, 16, and 32. The factors of a³ are 1, a, a², and a³.
12a²: The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of a² are 1, a, and a².

The largest common factor of both 32a³ and 12a² is 4a². Now, we factor out 4a² from both terms:

32a³ / 4a² = 8a
12a² / 4a² = 3

Therefore, the fully factored form of 32a³ + 12a² is 4a²(8a + 3). This means that if you multiply 4a² by (8a + 3), you will get the original expression 32a³ + 12a².

Factoring algebraic expressions is a fundamental skill in algebra. It allows us to simplify expressions, solve equations, and gain a deeper understanding of the relationships between variables. By mastering factoring, you’ll open doors to more advanced concepts in algebra and beyond.

Let’s break down how to find the factored form of 16x² + 8x + 32.

The factored form is 8(2x² + x + 4).

We can achieve this by finding the greatest common factor (GCF) of the terms in the expression. The GCF of 16, 8, and 32 is 8. We then factor out the 8 from each term:

16x² / 8 = 2x²
8x / 8 = x
32 / 8 = 4

This gives us the factored form: 8(2x² + x + 4).

Factoring out the GCF is a helpful technique for simplifying expressions and solving equations. When factoring out a GCF, we’re essentially dividing each term in the expression by the GCF, and then placing the GCF outside of parentheses.

Let me know if you’d like to explore other examples of factoring!

Let’s break down how to factor 3x + 24y.

Factoring means finding the greatest common factor (GCF) of the terms in an expression and rewriting the expression as a product. In this case, the GCF of 3x and 24y is 3. We can rewrite the expression as follows:

3x + 24y = 3(x + 8y)

Here’s how it works:

Finding the GCF: The greatest common factor (GCF) is the largest number that divides into both terms. 3 is the GCF of 3 and 24.
Factoring out the GCF: We divide each term in the original expression by the GCF (3). This gives us x and 8y.
Writing the factored form: The GCF (3) is placed outside the parentheses, and the results of the divisions (x + 8y) are placed inside the parentheses.

To check our work: we can distribute the 3 back into the parentheses. This should give us the original expression:

3(x + 8y) = 3x + 24y

Factoring is essentially the reverse of the distributive property. It’s a helpful skill in algebra, as it allows us to simplify expressions and solve equations.

The greatest common factor of 3x and 24y is 3.

Let’s break down why:

Finding the GCF: To find the greatest common factor (GCF) of two or more numbers (or terms in this case), we look for the largest number that divides into all of them evenly.
Factors of 3x: 1, 3, x, and 3x
Factors of 24y: 1, 2, 3, 4, 6, 8, 12, 24, y, 2y, 3y, 4y, 6y, 8y, 12y, and 24y

The largest number that appears in both lists is 3.

Factoring out the GCF: Now, we can rewrite the expression 3x + 24y by factoring out the GCF:

3x + 24y = 3(x + 8y)

* This means we’re essentially “pulling out” the common factor of 3 from both terms.

Let’s check our answer:

3(x + 8y) = 3x + 24y

We’ve successfully factored the expression!

In summary: Factoring is like finding the building blocks of an expression. We look for the greatest common factor (GCF), pull it out, and see what’s left. This helps us simplify expressions and understand their structure better.

Let’s break down factoring and explore how to find the factored form of 3x – 24.

Factoring is like finding the building blocks of an expression. We try to rewrite the expression as a product of simpler terms called factors.

To factor 3x – 24, we notice that both terms, 3x and 24, share a common factor: 3. We can pull out this common factor:

3x – 24 = 3(x – 8)

Here’s how it works:

3x divided by 3 gives us x.
24 divided by 3 gives us 8.

Therefore, the factored form of 3x – 24 is 3(x – 8).

Let’s dive a bit deeper into the concept of factoring:

Factoring is a crucial skill in algebra. It allows us to simplify expressions, solve equations, and gain a deeper understanding of the relationships between different terms.

In the case of 3x – 24, we were able to factor out a common numerical factor. This is a common type of factoring known as factoring out a greatest common factor. However, there are other techniques for factoring, such as difference of squares, perfect square trinomials, and grouping.

Here are some additional examples of factoring:

x² – 4 = (x + 2)(x – 2) (Difference of Squares)
x² + 6x + 9 = (x + 3)² (Perfect Square Trinomial)

Understanding how to factor expressions is essential for building a strong foundation in algebra. With practice, you’ll become more comfortable with various factoring techniques and their applications in solving problems.

Let’s break down 3x + 4y = 24. This is a linear equation, which means it represents a straight line when graphed. You can think of it like a recipe for drawing a line:

x and y represent the coordinates on a graph (think of them like the ingredients).
3 and 4 are the coefficients, which determine the slope and steepness of the line (imagine the amounts of each ingredient).
24 is the constant term, which determines where the line crosses the y axis (like the final touch on your recipe).

To graph this equation, you can find two points on the line and connect them. Here’s how we can find those points:

1. Finding the x-intercept: This is where the line crosses the x axis. On the x axis, y = 0. So, we can substitute 0 for y in the equation:

3x + 4(0) = 24
3x = 24
x = 8

This gives us the point (8, 0).

2. Finding the y-intercept: This is where the line crosses the y axis. On the y axis, x = 0. Let’s substitute 0 for x in the equation:

3(0) + 4y = 24
4y = 24
y = 6

This gives us the point (0, 6).

Now, you can plot these two points on a graph and draw a straight line through them. That line represents the equation 3x + 4y = 24.

It’s important to remember that this equation has infinitely many solutions. Every point that lies on that line satisfies the equation. We just found two easy points to help us draw the line.

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