Algebraic Technique - 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 $3a^3$, we recognize that $a^3$ means $a \times a \times a$. Therefore, $3a^3$ can be decomposed as $3 \cdot a\cdot a\cdot a$.

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:

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

To break down the expression $3y^2 + 6$, we need to recognize common factors or express terms in basic forms.

The term $3y^2$ can be rewritten by breaking down the operations: $3\cdot y\cdot y$.

The constant $6$ remains as it is in its basic term.

Thus, the broken down expression becomes $3\cdot y\cdot y + 6$.

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 $4 \cdot a \cdot a$.

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 break down the expression$4x^2 + 6x$ into its basic terms, we need to look for a common factor in both terms.

The first term is $4x^2$, which can be rewritten as $4\cdot x\cdot x$.

The second term is$6x$, which can be rewritten as $2\cdot 3\cdot x$.

The common factor between the terms is $x$.

Thus, the expression can be broken down into $4\cdot x^2 + 6\cdot x$, and further rewritten with common factors as $4\cdot x\cdot x + 6\cdot x$.

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 $5x^2 + 10$, identify the common factors.

The first term is $5x^2$, which can be rewritten as $5\cdot x\cdot x$.

The second term is $10$, which can be rewritten as $5\cdot 2$.

Notice that both terms share a common factor of $5$.

This allows the expression to be broken down to $5(x^2) + 10$, which translates to $5\cdot x\cdot x + 10$ using common terms.

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 $6b^2$ 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 $6 \cdot b \cdot b$.

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 $8y$, 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 $y$.