Problem 51 - Algebra for College Students

Question

Perform the indicated operations. $$\left(2 x^{2}-7 x-1\right)+\left(-4 x^{2}-x+6\right)+\left(-7 x^{2}-4 x-1\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-9x^2 - 12x + 4\).

Step 1: Expand the Expression

Write out the entire expression combining all the given polynomials: \[ (2x^2 - 7x - 1) + (-4x^2 - x + 6) + (-7x^2 - 4x - 1) \] This expression shows each polynomial that will be combined.

Step 2: Remove Parentheses

Remove the parentheses from each polynomial term since there are no multiplication signs affecting these parentheses. Simply write: \[ 2x^2 - 7x - 1 - 4x^2 - x + 6 - 7x^2 - 4x - 1 \]

Step 3: Group Like Terms

Identify and group together the like terms from the expression: Terms involving \(x^2\): \[ (2x^2 - 4x^2 - 7x^2) \]Terms involving \(x\): \[ (-7x - x - 4x) \]Constant terms: \[ (-1 + 6 - 1) \]

Step 4: Simplify Like Terms

Calculate the sum for each group of like terms: \(x^2\) terms: \[ 2x^2 - 4x^2 - 7x^2 = -9x^2 \]\(x\) terms: \[ -7x - x - 4x = -12x \]Constant terms: \[ -1 + 6 - 1 = 4 \]

Step 5: Combine Simplified Terms

Combine the simplified terms to write the final simplified expression: \[ -9x^2 - 12x + 4 \]

Key Concepts

Combining Like Terms

When working with polynomials, one essential skill is combining like terms. This means grouping together terms in a polynomial that have the same variable raised to the same power. For example, in the expression \(2x^2 - 7x - 1 - 4x^2 - x + 6 - 7x^2 - 4x - 1\), the terms \(2x^2\), \(-4x^2\), and \(-7x^2\) are like terms because they all involve \(x^2\). Similarly, terms such as \(-7x\), \(-x\), and \(-4x\) are grouped together because they each involve \(x\) to the first power.

• Like terms must have the same variables raised to the same power.
• Combining like terms simplifies expressions and helps in solving polynomial equations.

Once you've identified like terms, you can add or subtract the coefficients. For instance, combining the \(x^2\) terms in the expression results in \(2x^2 - 4x^2 - 7x^2 = -9x^2\). The same process can be applied to \(x\) terms and constant terms.

Polynomial Addition

Polynomial addition is similar to combining like terms because it involves adding polynomials by combining their corresponding terms. When adding polynomials, you effectively distribute each term across the others and then group like terms for simplification.
For example, when you add \( (2x^2 - 7x - 1) + (-4x^2 - x + 6) + (-7x^2 - 4x - 1) \), each term is combined based on its degree:

• Add all \(x^2\) terms together.
• Add all \(x\) terms together.
• Add all constant terms together.

This process helps to keep polynomials organized and eases further computation. Notice that the exact ordering of terms does not affect the ultimate outcome; it's about ensuring each degree of terms is appropriately managed through addition.

Simplifying Expressions

Simplifying expressions is a critical step in working with polynomials. This involves reducing the expression into its simplest form by removing unnecessary terms and combining like terms to create a concise representation. Simplification makes expressions easier to understand and solve.
Upon combining like terms in our example, we simplify \(-9x^2 - 12x + 4\). This polynomial is now much easier to handle for further operations—whether factoring, evaluating, or solving for specific variables.
By simplifying expressions:

• We reduce the complexity of the polynomial.
• We make further operations more straightforward.
• Our final expression becomes cleaner and easier to interpret.

Remember, the goal of simplifying is not just to "tidy up" the expression, but to prepare it for whatever mathematical manipulation comes next, be it solving or graphing.