Algebraic Technique - Examples, Exercises and Solutions

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 $y$.

Let's remember the extended distributive property:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$Note that the operation between the terms inside the parentheses is a multiplication operation:

$(a b)(c d)$Unlike in the extended distributive property previously mentioned, which is addition (or subtraction, which is actually the addition of the term with a minus sign),

Also, we notice that since there is a multiplication among all the terms, both inside the parentheses and between the parentheses, this is a simple multiplication and the parentheses are actually not necessary and can be remoed. We get:

$(a b)(c d)= \\ abcd$Therefore, opening the parentheses in the given expression using the extended distributive property is incorrect and produces an incorrect result.

Therefore, the correct answer is option d.

We simplify the given expression by opening the parentheses using the extended distributive property:

$(\textcolor{red}{x}+\textcolor{blue}{y})(t+d)=\textcolor{red}{x}t+\textcolor{red}{x}d+\textcolor{blue}{y}t+\textcolor{blue}{y}d$Keep in mind that in the distributive property formula mentioned above, we assume that the operation between the terms inside the parentheses is an addition operation, therefore, of course, we will not forget that the sign of the term's coefficient is ery important.

We will also apply the rules of multiplication of signs, so we can present any expression within parentheses that's opened with the distributive property as an expression with addition between all the terms.

In this expression we only have addition signs in parentheses, therefore we go directly to opening the parentheses,

$$We start by opening the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{c})(4+c)\\ \textcolor{red}{x}\cdot 4+\textcolor{red}{x}\cdot c+\textcolor{blue}{c}\cdot 4+\textcolor{blue}{c} \cdot c\\ 4x+xc+4c+c^2$To simplify this expression, we use the power law for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step like terms come into play.

We define like terms as terms in which the variables (in this case, x and c) have identical powers (in the absence of one of the variables from the expression, we will refer to its power as zero power, this is because raising any number to the power of zero results in 1).

We will also use the substitution property, and we will order the expression from the highest to the lowest power from left to right (we will refer to the regular integer as the power of zero),

Keep in mind that in this new expression there are four different terms, this is because there is not even one pair of terms in which the variables (different) have the same power. Also it is already ordered by power, therefore the expression we have is the final and most simplified expression:$\textcolor{purple}{4x}\textcolor{green}{+xc}\textcolor{black}{+4c}\textcolor{orange}{+c^2 }\\ \textcolor{orange}{c^2 }\textcolor{green}{+xc}\textcolor{purple}{+4x}\textcolor{black}{+4c}\\$We highlight the different terms using colors and, as emphasized before, we make sure that the main sign of the term is correct.

We use the substitution property for multiplication to note that the correct answer is option A.