Expanding and Simplifying Product of Two Binomials

Oct 13, 2025 by leolavanderia.com.br 80 views

In algebra, expanding and simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. In this article, we will explore how to expand and simplify six different algebraic expressions, each containing a product of two or more terms.

Expanding and Simplifying Product of Two Binomials

To expand and simplify a product of two binomials, we can use the distributive property, which states that for any real numbers a, b, c, and d:

a(b + c) = ab + ac

Using this property, we can expand and simplify each of the given expressions.

1. Expanding and Simplifying ${$ a(4a + 3) $}$

To expand and simplify the expression ${$ a(4a + 3) $}$, we can use the distributive property:

${$ a(4a + 3) $}$ = ${$ a \cdot 4a $}$ + ${$ a \cdot 3 $}$

= ${$ 4a^2 $}$ + ${$ 3a $}$

Therefore, the expanded and simplified form of the expression is ${$ 4a^2 + 3a $}$.

2. Expanding and Simplifying ${$ x(2x - 5) $}$

To expand and simplify the expression ${$ x(2x - 5) $}$, we can use the distributive property:

${$ x(2x - 5) $}$ = ${$ x \cdot 2x $}$ - ${$ x \cdot 5 $}$

= ${$ 2x^2 $}$ - ${$ 5x $}$

Therefore, the expanded and simplified form of the expression is ${$ 2x^2 - 5x $}$.

3. Expanding and Simplifying ${$ -3n(n^2 + 2n) $}$

To expand and simplify the expression ${$ -3n(n^2 + 2n) $}$, we can use the distributive property:

${$ -3n(n^2 + 2n) $}$ = ${$ -3n \cdot n^2 $}$ + ${$ -3n \cdot 2n $}$

= ${$ -3n^3 $}$ + ${$ -6n^2 $}$

Therefore, the expanded and simplified form of the expression is ${$ -3n^3 - 6n^2 $}$.

4. Expanding and Simplifying ${$ 3x(5x^2 - x + 4) $}$

To expand and simplify the expression ${$ 3x(5x^2 - x + 4) $}$, we can use the distributive property:

${$ 3x(5x^2 - x + 4) $}$ = ${$ 3x \cdot 5x^2 $}$ - ${$ 3x \cdot x $}$ + ${$ 3x \cdot 4 $}$

= ${$ 15x^3 $}$ - ${$ 3x^2 $}$ + ${$ 12x $}$

Therefore, the expanded and simplified form of the expression is ${$ 15x^3 - 3x^2 + 12x $}$.

5. Expanding and Simplifying ${$ -4b(1 - 9b - 2b^2) $}$

To expand and simplify the expression ${$ -4b(1 - 9b - 2b^2) $}$, we can use the distributive property:

${$ -4b(1 - 9b - 2b^2) $}$ = ${$ -4b \cdot 1 $}$ - ${$ -4b \cdot 9b $}$ - ${$ -4b \cdot 2b^2 $}$

= ${$ -4b $}$ + ${$ 36b^2 $}$ + ${$ 8b^3 $}$

Therefore, the expanded and simplified form of the expression is ${$ -4b + 36b^2 + 8b^3 $}$.

6. Expanding and Simplifying ${$ 2m^2(2m2 + 3m - 5) $}$

To expand and simplify the expression ${$ 2m^2(2m2 + 3m - 5) $}$, we can use the distributive property:

${$ 2m^2(2m2 + 3m - 5) $}$ = ${$ 2m^2 \cdot 2m^2 $}$ + ${$ 2m^2 \cdot 3m $}$ - ${$ 2m^2 \cdot 5 $}$

= ${$ 4m^4 $}$ + ${$ 6m^3 $}$ - ${$ 10m^2 $}$

Therefore, the expanded and simplified form of the expression is ${$ 4m^4 + 6m^3 - 10m^2 $}$.

In the previous article, we explored how to expand and simplify six different algebraic expressions, each containing a product of two or more terms. In this article, we will answer some frequently asked questions about expanding and simplifying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers a, b, c, and d:

a(b + c) = ab + ac

This property allows us to expand and simplify expressions containing products of two or more terms.

Q: How do I expand and simplify an expression containing a product of two binomials?

A: To expand and simplify an expression containing a product of two binomials, you can use the distributive property. For example, to expand and simplify the expression ${$ a(4a + 3) $}$, you can use the distributive property as follows:

${$ a(4a + 3) $}$ = ${$ a \cdot 4a $}$ + ${$ a \cdot 3 $}$

= ${$ 4a^2 $}$ + ${$ 3a $}$

Q: What is the difference between expanding and simplifying an expression?

A: Expanding an expression means multiplying out the terms, while simplifying an expression means combining like terms. For example, to expand and simplify the expression ${$ 2x(3x + 4) $}$, you can use the distributive property as follows:

${$ 2x(3x + 4) $}$ = ${$ 2x \cdot 3x $}$ + ${$ 2x \cdot 4 $}$

= ${$ 6x^2 $}$ + ${$ 8x $}$

To simplify the expression, you can combine like terms:

${$ 6x^2 + 8x $}$

Q: How do I simplify an expression containing a product of three or more terms?

A: To simplify an expression containing a product of three or more terms, you can use the distributive property and combine like terms. For example, to simplify the expression ${$ 3x(2x + 3y + 4) $}$, you can use the distributive property as follows:

${$ 3x(2x + 3y + 4) $}$ = ${$ 3x \cdot 2x $}$ + ${$ 3x \cdot 3y $}$ + ${$ 3x \cdot 4 $}$

= ${$ 6x^2 $}$ + ${$ 9xy $}$ + ${$ 12x $}$

To simplify the expression, you can combine like terms:

${$ 6x^2 + 9xy + 12x $}$

Q: What are some common mistakes to avoid when expanding and simplifying expressions?

A: Some common mistakes to avoid when expanding and simplifying expressions include:

Forgetting to use the distributive property when expanding an expression
Not combining like terms when simplifying an expression
Making errors when multiplying out the terms
Not checking the expression for any errors or simplifications

Q: How can I practice expanding and simplifying expressions?

A: You can practice expanding and simplifying expressions by working through examples and exercises in your textbook or online resources. You can also try creating your own expressions and challenging yourself to expand and simplify them.

Q: What are some real-world applications of expanding and simplifying expressions?

A: Expanding and simplifying expressions has many real-world applications, including:

Solving equations and inequalities in physics and engineering
Modeling population growth and decay in biology and economics
Analyzing data and making predictions in statistics and data science
Creating and solving mathematical models in finance and economics

In conclusion, expanding and simplifying algebraic expressions is an essential skill in mathematics that has many real-world applications. By understanding the distributive property and how to expand and simplify expressions, you can solve equations and manipulate mathematical statements with ease.