Problem 52 - Algebra for College Students

Question

Perform the indicated operations. $$\left(5 x^{2}+x+4\right)+\left(-x^{2}+2 x+4\right)+\left(-14 x^{2}-x+6\right)$$

Step-by-Step Solution

Answer

The result is \(-10x^2 + 2x + 14\).

Step 1: Distribute Connecting Signs

The exercise consists of adding three polynomial expressions. Start by distributing any connecting signs, if necessary. In this case, each polynomial is grouped with parentheses, but no signs need to be distributed since the expressions are separated only by addition signs.

Step 2: Identify Like Terms

Identify the 'like terms'—terms with the same variable raised to the same power—in the polynomial expressions. Here, the like terms are the terms with \(x^2\) (second degree), terms with \(x\) (first degree), and the constant terms.

Step 3: Add the x² Terms

Combine the coefficients of the \(x^2\) terms. The terms are \(5x^2\), \(-x^2\), and \(-14x^2\). So, compute 5 + (-1) + (-14) to get \(-10x^2\).

Step 4: Add the x Terms

Now combine the coefficients of the \(x\) terms. The terms are \(x\),\(2x\), and \(-x\). Compute 1 + 2 - 1 to get \(2x\).

Step 5: Add the Constant Terms

Finally, combine the constant terms. The constants are \(4\), \(4\), and \(6\). Calculate 4 + 4 + 6 to get 14.

Step 6: Write the Result

After combining the like terms, write the final expression in standard form (highest degree to lowest): \(-10x^2 + 2x + 14\).

Key Concepts

Like Terms

In the world of algebra, identifying like terms is crucial when performing operations with polynomials. Like terms are those terms in an expression that have the exact same variables raised to the same powers. In our exercise, we have three types of terms to consider:

• Terms involving \(x^2\)
• Terms involving \(x\)
• Constant terms (numbers without variables)

Recognizing like terms allows us to simplify the expression by combining them. For instance, terms like \(5x^2\) and \(-x^2\) are considered like terms because they both contain \(x\) raised to the power of 2. This step sets the stage for simplification, making complex equations easier to manage through addition or subtraction of coefficients.

Distributive Property

The distributive property is a key concept in algebra that allows you to multiply a single term by terms within parentheses efficiently and without altering the expression's value. Although this specific exercise involves only addition, understanding the distributive property is still fundamental.When adding polynomials, the property helps to determine how expressions should be handled when simplifying or expanding equations. In simpler terms, it allows us to correctly deal with operations like \(a(b + c) = ab + ac\). This rule ensures we distribute multiplication across terms.Here, we need not distribute sign changes because there's no subtraction involved at the outermost level. However, being familiar with this principle ensures we handle all potential multiplications and additions consistently.

Combining Coefficients

Combining coefficients is the final step when adding polynomials after identifying like terms. Coefficients are the numerical parts of the terms, such as the "5" in \(5x^2\) or the "-1" in \(-x^2\). By adding or subtracting these coefficients, you directly simplify the polynomial.In our exercise:

• We combined the coefficients of \(x^2\) terms: \(5 + (-1) + (-14) = -10\)
• We combined the coefficients of the \(x\) terms: \(1 + 2 - 1 = 2\)
• Finally, we combined the constant terms: \(4 + 4 + 6 = 14\)

The key is to carefully maintain the signs of each coefficient during this process. This ensures that the simplification maintains the original polynomial's value, often resulting in a new, simpler equation such as their combination \(-10x^2 + 2x + 14\). Remember, combining coefficients properly brings clarity and precision to polynomial operations.