Problem 50 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 5\left[3 x+7\left(2 x^{2}+3 x+2\right)+5\right]-10 x^{2}+4\left(3 x^{2}+x\right) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given algebraic expression is $$72 x^{2} + 124x + 95$$.

Step 1: Distribute the constants inside the parentheses

First, we will apply the distributive property by multiplying the constants outside the parentheses with the terms inside the parentheses. $$ 5[3 x+7(2 x^{2}+3 x+2)+5]-10 x^{2}+4(3x^{2}+x) \\ $$

Step 2: Simplify the expression

Now, we will simplify the expression by combining like terms. $$ 15x + 5\cdot7(2 x^{2}+3 x+2) + 25 -10 x^{2}+ 12x^{2} + 4x \\ $$

Step 3: Distribute the 35 inside the parentheses

We will now multiply the remaining constant (35) outside the parentheses with the terms inside the parentheses: $$ 15x + (70 x^{2}+105 x+70) + 25 -10 x^{2}+ 12x^{2} + 4x \\ $$

Step 4: Combine like terms

Finally, we will combine the like terms to obtain the simplified algebraic expression: $$ (70 x^{2} - 10 x^{2} + 12 x^{2}) + (15x + 105x + 4x) + (70 + 25) \\ $$ This becomes: $$ 72 x^{2} + 124x + 95 \\ $$ So the simplified algebraic expression is: $$ 72 x^{2} + 124x + 95 $$

Key Concepts

Distributive Property

When tackling algebraic expressions, the distributive property is a powerful tool. It allows us to simplify expressions by distributing multiplication over addition or subtraction inside parentheses. Mathematically, it can be expressed as: \[ a(b + c) = ab + ac \] Here, the term outside the parentheses is multiplied by each term inside. Consider our example, where we have: \[ 5[3x + 7(2x^{2} + 3x + 2) + 5] \] To begin, distribute 5 across the entire terms within the brackets. Following that, distribute each term within:

• Multiply 5 by \(3x\) to get \(15x\)
• Distribute 5 through \(7(2x^{2} + 3x + 2)\) after solving inside the second pair of parentheses
• 5 by 5 to get 25

Embracing the distributive property simplifies expressions and sets the foundation for further simplification.

Combining Like Terms

After using the distributive property, combining like terms is the next crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable parts raised to the same power, such as \(x^{2}\) terms or \(x\) terms.
In our exercise, we arrive at an expression like: \[ 70x^{2} - 10x^{2} + 12x^{2} + 15x + 105x + 4x + 70 + 25 \] Here, your goal is to identify and group like terms:

• Combine the \(x^{2}\) terms: \(70x^{2} - 10x^{2} + 12x^{2} = 72x^{2}\)
• Combine the \(x\) terms: \(15x + 105x + 4x = 124x\)
• Add the constant terms: \(70 + 25 = 95\)

By combining like terms, we condense the expression, making it easier to manage and understand. This step is crucial for arriving at the simplest form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent mathematical statements without an equal sign, differentiating them from equations. An algebraic expression can be a single number, a variable, or a combination of several elements connected by operators like addition, subtraction, multiplication, and division.
Consider the original problem: \[ 5[3x + 7(2x^{2} + 3x + 2) + 5] - 10x^{2} + 4(3x^{2} + x) \] This expression includes several components:

• Constants, such as 10 and 4.
• Variables like \(x^{2}\) and \(x\).
• Operators connecting these elements.

The task of simplifying the expression involves systematically applying mathematical principles such as the distributive property and like terms to reduce it to a manageable form. Understanding algebraic expressions is essential, as they form the backbone of algebra, enabling us to model real-world situations and solve problems.