Problem 8 - Elementary Algebra

Question

Simplify each of the following expressions by using the distributive property and combining like terms. $$ 7\left(x+x^{3}\right)-4 x^{3}-x+1+4\left(x^{2}-2 x^{3}+7\right) $$

Step-by-Step Solution

Answer

Question: Simplify the expression using the distributive property and combining like terms: $7(x+x^3) - 4x^3 - x + 1 + 4(x^2 - 2x^3 + 7)$. Answer: The simplified expression is $6x - 5x^3 + 4x^2 + 29$.

Step 1: Expand the expression using the distributive property

To start simplifying the expression, we first need to expand the expression by applying the distributive property. This means that we will multiply the terms inside the parentheses by the terms outside of the parentheses: $$ 7\left(x+x^{3}\right)-4 x^{3}-x+1+4\left(x^{2}-2 x^{3}+7\right) \\ = 7x + 7x^{3} - 4x^{3} - x + 1 + 4x^{2} - 8x^{3} + 28 $$

Step 2: Combine like terms

Now, we can simplify the expression by combining the like terms. This means we will group the terms with the same power of x and add/subtract them: $$ = (7x - x) + (7x^{3} - 4x^{3} - 8x^{3}) + 4x^{2} + (1 + 28) \\ = 6x - 5x^{3} + 4x^{2} + 29 $$ So the simplified expression is: $$ 6x - 5x^{3} + 4x^{2} + 29 $$

Key Concepts

Combining like terms

Combining like terms is an essential step in simplifying algebraic expressions. When you simplify an expression, you look for terms that have the same variable raised to the same power. These are called 'like terms'. Once you identify these, you can add or subtract the coefficients of these terms to combine them.

• Look for each group of like terms, such as all terms with x, x², x³, or constant numbers.
• Only combine terms with the same variable and the same exponent. For example, 2x and 5x are like terms, but 2x and 2x² are not.
• Add or subtract the coefficients of the like terms. For example, in the problem, (7x - x) becomes 6x because you subtract 1x from 7x.

Combining like terms simplifies expressions by reducing them to the fewest number of terms possible, making them easier to handle in further calculations.

Simplifying expressions

Simplifying expressions involves reducing an algebraic expression to its simplest form. This means minimizing the number of terms, getting rid of parentheses through distribution, and combining like terms.
The process often involves the following steps:

• Distribute terms across parentheses. For example, in the original problem, we distribute 7 over (x + x³), resulting in (7x + 7x³).
• Combine like terms, as previously discussed, to further reduce the expression into fewer terms.
• Arrange the terms in a conventional order, typically from highest to lowest power of the variable (e.g., x³ comes before x² which comes before x).

Simplifying helps in understanding an algebraic expression better and prepares you for solving equations if that step is necessary.

Algebraic expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations. They do not include an equal sign—which differentiates them from algebraic equations. Algebraic expressions can look complex but can often be simplified through different mathematical processes like the distributive property and combining like terms.

• The basic components of an expression are constants (fixed numbers), variables (symbols, usually letters like x), and coefficients (numbers multiplying the variables).
• Expressions can include addition, subtraction, multiplication, and division operations.
• Common processes for simplifying these expressions include using the distributive property to expand brackets and then combining like terms.

Understanding how to manipulate algebraic expressions is foundational for moving on to solving algebraic equations and further algebraic concepts.