Simplifying Algebraic Expressions: Videos & Practice Problems

The distributive property is a fundamental algebraic principle that connects multiplication and addition or subtraction within parentheses. It states that when a number or variable multiplies a sum or difference inside parentheses, you can distribute the multiplication to each term individually. Mathematically, this is expressed as \(a \times (b + c) = a \times b + a \times c\). This property also applies to subtraction, since subtraction can be viewed as adding a negative number, so \(a \times (b - c) = a \times b - a \times c\).

For example, consider the expression \(2 \times (4 + 5)\). Using the order of operations, you first add inside the parentheses: \(4 + 5 = 9\), then multiply: \(2 \times 9 = 18\). Alternatively, applying the distributive property, you multiply 2 by each term inside the parentheses: \(2 \times 4 + 2 \times 5 = 8 + 10 = 18\). Both methods yield the same result, confirming the validity of the distributive property.

This property becomes especially useful when dealing with variables, where addition or subtraction inside parentheses cannot be simplified directly. For instance, in the expression \(4 \times (x - 8)\), since \(x\) is unknown, you cannot simplify inside the parentheses first. Instead, distribute the 4: \(4 \times x - 4 \times 8 = 4x - 32\). This allows you to rewrite expressions without parentheses, facilitating further algebraic manipulation.

The distributive property also works when the multiplier is on the right side of the parentheses, due to the commutative property of multiplication (i.e., \(a \times b = b \times a\)). For example, in \( (4x + 2y - 7z) \times 3\), you distribute the 3 to each term inside the parentheses: \(3 \times 4x + 3 \times 2y - 3 \times 7z = 12x + 6y - 21z\). This demonstrates that the distributive property applies regardless of the number of terms inside the parentheses or the position of the multiplier.

Understanding and applying the distributive property is essential for simplifying algebraic expressions, solving equations, and working with variables efficiently. It bridges the operations of multiplication and addition/subtraction, enabling the expansion and simplification of expressions in a consistent and reliable way.

When simplifying an expression with a negative sign outside the parentheses, such as −(2x − 3y + 4 − z), the negative sign acts as a multiplier of −1 applied to every term inside the parentheses. This means you distribute −1 across each term: −1 × 2x, −1 × (−3y), −1 × 4, and −1 × (−z). It is important to carefully handle the signs during this process, especially when subtraction is involved inside the parentheses.

Applying the distributive property, the multiplication results in:

\(-1 \times 2x = -2x\)
\(-1 \times (-3y) = +3y\)
\(-1 \times 4 = -4\)
\(-1 \times (-z) = +z\)

Thus, the simplified expression becomes:

\(-2x + 3y - 4 + z\)

This final expression contains the same terms as the original but with opposite signs, which is the expected outcome when multiplying by a negative number. Recognizing this pattern can save time by allowing you to directly change the signs of each term without performing each multiplication step. However, if there is any uncertainty, fully applying the distributive property ensures accuracy.

The distributive property allows us to rewrite expressions by factoring out a common factor and enclosing the remaining terms in parentheses. Instead of expanding expressions by distributing multiplication over addition or subtraction, we can reverse this process to simplify or factor expressions.

For example, consider the expression 91 × m + 91 × n. Both terms share the common factor 91. By factoring out 91, we rewrite the expression as 91 × (m + n). This works because distributing 91 back over m + n returns the original expression, confirming the equivalence.

Similarly, for the expression 4 × x − 4 × y, the common factor is 4. Factoring out 4 gives 4 × (x − y). Distributing 4 back over x − y yields the original expression, demonstrating the correctness of the factoring.

This process of factoring by applying the distributive property in reverse is fundamental in algebra. It helps simplify expressions and solve equations more efficiently. Recognizing common factors and rewriting expressions with parentheses enhances problem-solving skills and prepares for more advanced topics such as polynomial factoring and equation solving.

Algebraic expressions consist of numbers and variables combined through operations such as addition and subtraction. The individual parts of an expression separated by plus or minus signs are called terms. For example, in the expression 2x + 5, the terms are 2x and 5.

Like terms are terms that contain the exact same variables raised to the same exponents. The order of variables does not affect whether terms are like terms because multiplication is commutative. For instance, 4x² and 7x² are like terms since both have the variable x raised to the power of 2. Similarly, 2ab and 8ba are like terms because they contain the same variables a and b, each raised to the first power. Another example is 5x²yz and -10x²yz, where all variables and their exponents match exactly.

It is important to note that coefficients—the numerical factors in terms—do not determine whether terms are like terms. Terms are like terms solely based on their variables and exponents.

On the other hand, unlike terms differ either in their variables or in the exponents of those variables. For example, 4x² and 7x³ are not like terms because the exponents of x differ. Terms like 2a and 2ba are unlike because they contain different variables. Also, 5x²yz and -10xy²z are unlike terms since the exponents of x and y differ between the two.

Combining like terms is a fundamental skill in algebra. This process involves adding or subtracting the coefficients of like terms while keeping the variable part unchanged. For example, combining 4x² + 7x² results in 11x² because the coefficients 4 and 7 are added. Similarly, 2ab + 8ba can be combined as 10ab after recognizing the terms as like terms and rearranging variables if needed.

It is crucial to remember that unlike terms cannot be combined by adding or subtracting their coefficients. For example, 5x²yz - 10xy²z cannot be simplified further because the terms are not like terms due to differing exponents.

Understanding how to identify terms, recognize like terms, and combine them correctly is essential for simplifying algebraic expressions and solving equations efficiently.

When combining like terms in algebraic expressions, it is essential to identify terms that have the same variable raised to the same exponent. For example, in the expression 6a² + 8a², both terms contain the variable a raised to the power of 2. Since these are like terms, their coefficients can be added together, resulting in 14a². This process involves simply adding the numerical coefficients while keeping the variable and its exponent unchanged.

In contrast, consider the expression 7x² + 8x + 4. Although the first two terms share the variable x, the exponents differ: the first term has x² (x squared), while the second term has x to the first power. Because the exponents are not the same, these terms are not like terms and cannot be combined. The constant term 4 stands alone without any like terms to combine with, so the expression remains as is.

Another example is the expression -5y⁵ + 3y⁵. Both terms have the variable y raised to the fifth power, making them like terms. Combining their coefficients, -5 + 3, yields -2y⁵. This demonstrates that when combining like terms, the coefficients are added algebraically, taking into account their signs, while the variable and exponent remain unchanged.

Understanding how to combine like terms is fundamental in simplifying algebraic expressions. It requires recognizing terms with identical variables and exponents and then performing addition or subtraction on their coefficients. This skill lays the groundwork for more advanced algebraic manipulations and problem-solving.

When simplifying algebraic expressions, it is important to combine like terms so that each variable appears only once. Like terms are terms that have the same variable raised to the same exponent. For example, in the expression 3p - 9p + 4 - 2 + p, the terms 3p, -9p, and p are like terms because they all contain the variable p raised to the first power (an invisible exponent of 1).

To combine these like terms, add or subtract their coefficients. The coefficients here are 3, -9, and 1 (since p is the same as 1p). Calculating the sum: \(3 - 9 + 1 = -5\), so the combined term is \(-5p\).

Next, consider the constants in the expression, which are numbers without variables. The constants 4 and -2 are also like terms because they share the same variable—none. Adding these constants gives \(4 - 2 = 2\).

After combining all like terms, the simplified expression is \(-5p + 2\). This process of combining like terms is fundamental in algebra for simplifying expressions and solving equations efficiently.

When simplifying algebraic expressions, it is important to combine like terms so that each variable appears only once. Consider the expression 6z - 4 + 11z + 9. To identify like terms, look for terms that have the same variable raised to the same exponent. Here, both 6z and 11z contain the variable z with an implied exponent of 1, making them like terms. These can be combined by adding their coefficients: \(6 + 11 = 17\), resulting in \$17z\(.

Next, focus on the constant terms, which are numbers without variables. In this expression, the constants are \)-4\( and \(9\). Since they are like terms (both constants), they can be combined by simple addition: \)-4 + 9 = 5$.

After combining like terms, the simplified expression is \(17z + 5\). This expression has the variable z appearing only once and a single constant term, making it fully simplified. Understanding how to combine like terms is fundamental in algebra for simplifying expressions and solving equations efficiently.

To simplify algebraic expressions effectively, it is essential to understand that a fully simplified expression contains no parentheses and has all like terms combined. The process begins by distributing any constants or variables across parentheses. For example, when you have an expression like \(-z + 2(3 + 5z)\), distribute the 2 to both 3 and \$5z\(, resulting in \)-z + 6 + 10z\(. This step ensures that all terms are expressed without parentheses, making it easier to identify like terms.

Like terms are terms that share the same variable raised to the same exponent. In the expression above, \)-z\( and \)10z\( are like terms because both contain the variable \)z\( to the first power. Constants, such as 6, are not like terms with variables and remain separate. Grouping like terms by writing them next to each other, such as \)-z + 10z + 6\(, prepares the expression for combining.

Combining like terms involves adding or subtracting their coefficients. Since \)-z\( is equivalent to \)-1z\(, adding \)-1z + 10z\( yields \)9z\(. Thus, the simplified expression becomes \(9z + 6\). This process reduces the complexity of the original expression, making it more concise and easier to work with.

Consider another example: \(2a^2 - (6a^2 - b^2) + 5b^2\). First, distribute the negative sign (which is equivalent to multiplying by \)-1\() across the parentheses, changing the expression to \(2a^2 - 6a^2 + b^2 + 5b^2\). Next, identify like terms: \)2a^2\( and \)-6a^2\( are like terms because they both contain \)a^2\(, and \)b^2\( and \)5b^2\( are like terms as well.

Grouping these like terms together, we have \)(2a^2 - 6a^2) + (b^2 + 5b^2)\(. Combining the coefficients, \(2 - 6\) equals \)-4\(, so the first group simplifies to \)-4a^2\(. For the second group, \(1 + 5\) equals \(6\), so it simplifies to \)6b^2\(. The fully simplified expression is therefore \)-4a^2 + 6b^2$.

By following these steps—distributing constants or variables, identifying and grouping like terms, and combining coefficients—you can simplify any algebraic expression. This method not only streamlines expressions but also lays a strong foundation for solving more complex algebraic problems.

To simplify an algebraic expression by expanding and combining like terms, the first step involves distributing any constants or variables across the terms inside parentheses. For example, when distributing 3x across the terms 2x² and -5xy, multiply the coefficients and then apply the laws of exponents to the variables. Multiplying 3x by 2x² results in 6x³ because 3 times 2 equals 6, and x times x² equals x³. Similarly, distributing 3x to -5xy gives -15x²y, since 3 times -5 is -15, and x times x is x², with y remaining as is.

Next, distribute -4y across x² and -2xy. Multiplying -4y by x² yields -4x²y. When multiplying -4y by -2xy, the product is 8xy² because the negatives cancel out, and y times y becomes y². It is important to maintain variables in alphabetical order for clarity, so terms are written as x²y rather than yx².

After distributing, identify like terms by comparing variables and their exponents. Terms such as -15x²y and -4x²y are like terms because they contain the same variables raised to the same powers. However, terms like 6x³ and 8xy² do not have like terms to combine with, as their variable exponents differ.

Combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged. For instance, -15x²y and -4x²y combine to form -19x²y. The final simplified expression after distribution and combination is:

\[6x^{3} - 19x^{2}y + 8xy^{2}\]

This process highlights the importance of careful distribution, attention to signs, and proper identification of like terms to simplify algebraic expressions effectively.