Problem 30 - Elementary Algebra

Question

\(h^{2}-8 h-7+2 h^{2}-4 h+3\)

Step-by-Step Solution

Answer

The simplified expression is \(3h^2 - 12h - 4\).

Step 1: Combine like terms

Identify and combine the like terms in the expression: - \(h^2\) terms: \(h^2\) and \(2h^2\) - \(h\) terms: \(-8h\) and \(-4h\) - Constants: -7 and 3

Step 2: Add the coefficients of the like terms

Combine the coefficients of the like terms:- For \(h^2\): \(1h^2 + 2h^2 = 3h^2\)- For \(h\): \(-8h - 4h = -12h\)- For constants: \(-7 + 3 =\)

Step 3: Write the simplified expression

Combine the results from Step 2 to form the simplified expression: The simplified expression is \(3h^2 - 12h - 4\).

Key Concepts

Like Terms

Here's how you can identify like terms:

• Look for terms that have the same variable and exponent.
• Ignore the coefficients (numbers in front of the variables) at first; focus on whether the variables and their exponents match.

By combining like terms, you simplify the expression, making further calculations much easier.

Coefficients

Coefficients are the numbers in front of the variables in algebraic expressions. They tell us how many times to multiply the variable.
For instance, in the term \(2h^2\), 2 is the coefficient and \(h^2\) is the variable part.
The coefficients give us information about the size or quantity of each variable term.

• When combining like terms, you simply add or subtract the coefficients while keeping the variable part the same.
• For example, combining \(h^2\) and \(2h^2\): the coefficients (1 and 2) are added, resulting in \(3h^2\).
• Similarly, for \(-8h\) and \(-4h\), the coefficients are combined as \(-8 + (-4) = -12\), giving \(-12h\).

Remember, coefficients are essential in determining the final simplified expression of an algebraic equation.

Polynomials

A polynomial is a mathematical expression that includes a sum of multiple terms, which can consist of constants, variables, and exponents.
In our exercise, the expression \(h^2 - 8h - 7 + 2h^2 - 4h + 3\) is a polynomial.
Here are some key characteristics of polynomials:

• They are made up of terms that are added or subtracted.
• Each term has a variable raised to a non-negative integer exponent.

Let's look at the structure:

• A polynomial can have multiple terms: constants (like -7 and 3), linear terms (like -8h and -4h), and quadratic terms (like \(h^2\) and \(2h^2\).
• Polynomials are usually written in order of descending powers of the variable (from the highest exponent to the lowest).

Simplifying a polynomial through combining like terms helpfully reduces its complexity, making it easier to work with. Finally, our simplified expression is \(3h^2 - 12h - 4\).