Factorization: Common factor extraction - Examples, Exercises and Solutions

The expression $2x^2$ can be factored and broken down into the following basic terms:

• The coefficient $2$ remains as it is since it is already a basic term.
• The term $x^2$ can be broken down into $x \cdot x$.
• Therefore, the entire expression can be written as $2 \cdot x \cdot x$.

This breakdown helps in understanding the multiplicative nature of the expression.

Among the provided choices, the correct one that matches this breakdown is choice 2: $2\cdot x\cdot x$.

To solve this problem, we'll clearly delineate the expression $6x$ as follows:

• The number 6 is the coefficient.
• The letter $x$ is the variable.
• These two components are connected by multiplication, represented as $6 \cdot x$.

Thus, the expression $6x$ is equivalent to $6 \cdot x$, where 6 is multiplied by $x$.

Examining the choice options:

• Choice 1: $6\cdot x$ is correct because it represents the expression as a product of the coefficient and the variable.
• Choice 2: $x \cdot x \cdot x \cdot x \cdot x \cdot x$ represents a repeated multiplication of $x$, not applicable here.
• Choice 3: $\frac{6}{x}$ represents division, not the required approach.
• Choice 4: Incorrect, as $6x$ can indeed be expressed as $6 \cdot x$.

Therefore, the best breakdown of the expression is $6 \cdot x$, matching choice 1.

To break down the expression $8y^2$, we identify the basic components. The expression $y^2$ is a shorthand for$y \times y$. Therefore, $8y^2$ can be decomposed as $8 \cdot y \cdot y$.

To break down the expression $3a^3$, we recognize that $a^3$ means $a \times a \times a$. Therefore, $3a^3$ can be decomposed as $3 \cdot a\cdot a\cdot a$.

To break down the expression $8y$, we can see it as the multiplication of $8$ and $y$:

$8y = 8 \cdot y$

This shows the expression as a product of two factors, $8$ and $y$.

To break down the expression $5m$, we recognize it as the product of $5$ and $m$:

$5m = 5 \cdot m$

This expression can be seen as a multiplication of the constant $5$ and the variable $m$.

To break down the expression $5x^2$ into its basic terms, we identify each component in the expression:

$5 \, \text{is a constant multiplier}$

$x^2$ means $x \cdot x$

Therefore, $5x^2$ can be rewritten as $5 \cdot x \cdot x$.

To break down the expression $3y^3$ into its basic terms, we understand the components of the expression:

$3 \, \text{is a constant multiplier}$

$y^3$ can be rewritten as $y \cdot y \cdot y$

Thus, $3y^3$ can be decomposed into $3 \cdot y \cdot y \cdot y$.

To break down the expression $4a^2$ into basic terms, we need to look at each factor:

$4 \, \text{is a constant multiplier}$

$a^2$ means $a \cdot a$

Hence, $4a^2$ is equivalent to $4 \cdot a \cdot a$.

To break down the expression $6b^2$ into its fundamental parts, we analyze each element:

$6 \, \text{is a constant multiplier}$

$b^2$ represents $b \cdot b$

Therefore, $6b^2$ is decomposed as $6 \cdot b \cdot b$.

The expression can be broken down as follows:

$3x^2 + 2x$

Breaking down each term we have:

- $3x^2$ becomes $3\cdot x\cdot x$

- $2x$ remains $2 \cdot x$

Finally, the expression is:

$3\cdot x\cdot x+2\cdot x$

The expression can be broken down as follows:

$4x^2 + 3x$

1. Notice that both terms contain a common factor of $x$.

2. Factor out the common $x$:

$x(4x + 3)$.

3. Thus, breaking down each term we have:

- $4x^2$ becomes $4x \cdot x$ after factoring out $x$.

- $3x$ remains $3 \cdot x$ after factoring out $x$.

Finally, the expression is:

$4x\cdot x + 3\cdot x$

To simplify the expression $5x^3 + 3x^2$, we can break it down into basic terms:

The term $5x^3$ can be written as $5 \cdot x \cdot x \cdot x$.

The term$3x^2$ can be written as $3 \cdot x \cdot x$.

Thus, the expression simplifies to$5 \cdot x \cdot x \cdot x + 3 \cdot x \cdot x$.

To rewrite the expression $6z^2 + z$, break it down into basic components:

The term $6z^2$ can be expressed as $6 \cdot z \cdot z$.

The term $z$ is $1 \cdot z$.

Thus, rewriting gives $6 \cdot z \cdot z + 1 \cdot z$.

To rewrite the expression $8x^2 - 4x$ using its basic components, we'll follow these steps:

• Step 1: Identify the greatest common factor of the terms.
• Step 2: Factor each term using the greatest common factor.

Let's go through each step:

Step 1: Recognize that both terms $8x^2$ and $4x$ contain $x$ as a common factor.
Moreover, the numerical coefficients 8 and 4 have a common factor of 4.

Step 2: Factor the expression:
- $8x^2$ can be expressed as $8 \cdot x \cdot x$.
- $4x$ can be written as $4 \cdot x$.

Bringing them together, we can rewrite the expression:

$8x^2 - 4x = 8 \cdot x \cdot x - 4 \cdot x$.

Thus, the solution to the problem is $8\cdot x\cdot x-4\cdot x$.