The mathematical expression 30×6 + 25×4 + 20×9 presents an opportunity to explore algebraic concepts and simplify the expression to its most concise form. By employing fundamental principles of algebra, we can break down the expression into manageable components and arrive at its result.

To begin, let’s simplify each term within the expression. Using the distributive property, we can factor out the common factor of x from the first two terms: 30×6 + 25×4 + 20×9 = 5x(6x) + 5x(4) + 4x(5). This factoring reveals that the expression has three common factors: 5, x, and 4.

Next, we can combine the simplified terms: 5x(6x) + 5x(4) + 4x(5) = 5x(6x + 4) + 4x(5). Further simplification yields 5x(10x) + 4x(5) = 50x^{2} + 20x.

## Factors of 30×6 + 25×4 + 20×9

### Common Factors

The common factors of 30×6 + 25×4 + 20×9 are:

- 5
- x
- 4

### Prime Factors

The prime factors of 30×6 + 25×4 + 20×9 are:

- 5
^{2} - x
^{2} - 2
^{2}

## Simplifying 30×6 + 25×4 + 20×9

To simplify 30×6 + 25×4 + 20×9, we can use the following steps:

- Factor out the greatest common factor (GCF) from each term. The GCF of 30×6, 25×4, and 20×9 is 5x.
- Simplify each term by dividing by the GCF. We get 6x, 4, and 5x.
- Combine like terms. We get 5x(6x + 4 + 5x) = 5x(11x + 4).

## 30×6 + 25×4 + 20×9 in Expanded Form

The expanded form of 30×6 + 25×4 + 20×9 is:

30×6 + 25×4 + 20×9 = 180x + 100x + 180x = 460x

## 30×6 + 25×4 + 20×9 in Factored Form

The factored form of 30×6 + 25×4 + 20×9 is:

30×6 + 25×4 + 20×9 = 5x(6x + 4 + 5x)

= 5x(11x + 4)

## 30×6 + 25×4 + 20×9 as a Product of Primes

30×6 + 25×4 + 20×9 can be expressed as a product of primes as follows:

30×6 + 25×4 + 20×9 = 5^{2}x^{2}(2^{2} + 1^{2} + 2^{2}x^{2})

= 5^{2}x^{2}(1 + 4x^{2})

## 30×6 + 25×4 + 20×9 as a Sum of Squares

30×6 + 25×4 + 20×9 can be expressed as a sum of squares as follows:

30×6 + 25×4 + 20×9 = (5x)^{2} + (4)^{2} + (5x)^{2}

= 25x^{2} + 16 + 25x^{2}

= 50x^{2} + 16

## 30×6 + 25×4 + 20×9 in Standard Form

The standard form of 30×6 + 25×4 + 20×9 is:

30×6 + 25×4 + 20×9 = 50x^{2} + 20x

## 30×6 + 25×4 + 20×9 with Integer Coefficients

The expression 30×6 + 25×4 + 20×9 has integer coefficients. This means that each coefficient is an integer (whole number). The integer coefficients of the expression are 30, 25, and 20.

## 30×6 + 25×4 + 20×9 with Rational Coefficients

The expression 30×6 + 25×4 + 20×9 can also be expressed with rational coefficients. Rational coefficients are coefficients that can be expressed as a fraction of two integers. The rational coefficients of the expression are 15/2, 5/2, and 10/3.

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