Which expression is equivalent to 10x2y + 25x2

The expression $ 10x^2y + 25x^2 $ is equivalent to $ \boxed{5x^2 (2y + 5)} $.

## Equivalent Expression To find the expression equivalent to $ 10x^2y + 25x^2 $, we perform factorization. ### Step-by-Step Factorization 1. **Identify the common factor** in both terms, which is $ 5x^2 $. 2. **Factor out** the greatest common factor (GCF) from the expression: $$ 10x^2y + 25x^2 = 5x^2(2y) + 5x^2(5) $$ 3. Combine the common factor: $$ 5x^2 (2y + 5) $$ Thus, the expression equivalent to $ 10x^2y + 25x^2 $ is: $$ \boxed{5x^2 (2y + 5)} $$ Where: - $10x^2y + 25x^2$ is the original expression - $5x^2 (2y + 5)$ is the factored form

15 Aug, 2024

· Mathematics

$5x^2(2y + 5)$
$5x^2y(5 + 20y)$
$10xy(x + 15y)$
$10x^2(y + 25)$

Short Answer

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Long Explanation

Explanation

Equivalent Expression

To find the expression equivalent to $10x^2y + 25x^2$ , we perform factorization.

Step-by-Step Factorization

Identify the common factor in both terms, which is $5x^2$ .
Factor out the greatest common factor (GCF) from the expression:

10x^2y + 25x^2 = 5x^2(2y) + 5x^2(5)

Combine the common factor:

5x^2 (2y + 5)

Thus, the expression equivalent to $10x^2y + 25x^2$ is:

\boxed{5x^2 (2y + 5)}

Where:

$10x^2y + 25x^2$ is the original expression
$5x^2 (2y + 5)$ is the factored form

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Emily Rosen

Mathematics Content Writer at Math AI

Emily Rosen is a recent graduate with a Master's in Mathematics from the University of Otago. She has been tutoring math students and working as a part-time contract writer for the past three years. She is passionate about helping students overcome their fear of math through easily digestible content.

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Concept

Greatest Common Factor

Definition

The greatest common factor (GCF) is the largest integer that divides two or more given integers without leaving a remainder. This means it is a shared factor of the given numbers. The GCF is also known as the greatest common divisor (GCD) or the highest common factor (HCF).

Finding the GCF

There are multiple methods to find the GCF of given numbers, including:

Prime Factorization
Euclidean Algorithm
Listing Factors

Prime Factorization

This method involves breaking down each number into its prime factors and then identifying the common prime factors.

For example, to find the GCF of 48 and 60:

Prime factorization of 48:
$48 = 2^4 \times 3$
Prime factorization of 60:
$60 = 2^2 \times 3 \times 5$
Identify the common factors and their lowest powers:
$\text{Common prime factors: } 2^2 \times 3 = 12$

So, the GCF of 48 and 60 is 12.

Euclidean Algorithm

This efficient method uses the principle that the GCD of two numbers is the same as the GCD of the smaller number and the remainder of the division of the larger number by the smaller number. The steps are repeated until the remainder is zero.

To find the GCF of 48 and 60 using the Euclidean Algorithm:

Divide 60 by 48 and find the remainder: $60 = 48 \times 1 + 12$
Replace 60 with 48 and 48 with the remainder 12, then repeat: $48 = 12 \times 4 + 0$

When the remainder is zero, the divisor at that stage is the GCF. Therefore, the GCF is 12.

Listing Factors

This method involves listing all the factors of the given numbers and identifying the largest common factor.

For example, consider the numbers 48 and 60:

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The common factors are 1, 2, 3, 4, 6, and 12. Hence, the largest common factor is 12.

Concept

Factoring Expressions

Explanation

Factoring expressions is a fundamental process in algebra that involves breaking down complex expressions into simpler, multiplicative components. This makes it easier to solve equations, simplify expressions, and understand the properties of mathematical functions.

Basic Concept

In algebra, a polynomial expression can be factored into products of simpler polynomials. For instance, consider the polynomial:

ax^2 + bx + c

This quadratic expression can often be factored into two binomials:

(ax + m)(bx + n)

Where $m$ and $n$ are numbers that satisfy certain conditions derived from the original coefficients $a$ , $b$ , and $c$ .

Factoring Techniques

There are several common techniques for factoring expressions, including:

Factoring Out the Greatest Common Factor (GCF): Identifying the largest factor that is present in each term of the polynomial.

6x^2 + 9x = 3x(2x + 3)

Factoring by Grouping: Combining terms with common factors and then factoring out the GCF from each group.

ax + ay + bx + by = a(x + y) + b(x + y) =

= (a + b)(x + y)

Difference of Squares: Recognizing expressions that fit the form $a^2 - b^2$ and factoring them as $(a - b)(a + b)$ .

x^2 - 9 = (x - 3)(x + 3)

Trinomials: Factoring expressions of the form $ax^2 + bx + c$ .

x^2 + 5x + 6 = (x + 2)(x + 3)

Importance of Factoring

Simplification: Factoring helps to simplify complex expressions, making them easier to work with.

Finding Roots: Factoring is often used to find the roots of equations, as the factored form makes it easier to set each component to zero and solve for the variable.

Solving Equations: Once an expression is factored, solving equations becomes a more straightforward process.

Example

Consider the polynomial:

x^2 - 5x + 6

To factor this, we look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, we have:

x^2 - 5x + 6 = (x - 2)(x - 3)

In conclusion, the purpose of factoring expressions is to breakdown complex polynomial expressions into simpler factors, which can then be used to simplify, solve, and analyze various algebraic problems. Understanding this fundamental concept is crucial for progressing in mathematics.