Problem 6 - Elementary and Intermediate Algebra

Question

Determine whether each pair of polynomials are opposites. Write yes or no. a. \(y+7\) and \(y-7\) b. \(b-20\) and \(20-b\) c. \(x^{2}+2 x-1\) and \(-x^{2}-2 x-1\)

Step-by-Step Solution

Answer

a. No, b. Yes, c. No

Step 1: Understand opposites

Two polynomials are opposites if adding them results in zero. To check, we will sum each pair of given polynomials and see if it equals zero.

Step 2: Check opposites for a

Add the polynomials \(y+7\) and \(y-7\): \[(y+7) + (y-7) = y + 7 + y - 7 = 2y\]Since the sum is \(2y\), not zero, they are not opposites.

Step 3: Check opposites for b

Add the polynomials \(b-20\) and \(20-b\):\[(b-20) + (20-b) = b - 20 + 20 - b = 0\]The result is zero, so these polynomials are opposites.

Step 4: Check opposites for c

Add the polynomials \(x^{2}+2x-1\) and \(-x^{2}-2x-1\):\[ (x^{2} + 2x - 1) + (-x^{2} - 2x - 1) = x^{2} + 2x - 1 - x^{2} - 2x - 1 = -2 \]The sum is \(-2\), not zero, so these polynomials are not opposites.

Key Concepts

Opposite Polynomials

The concept of opposite polynomials plays a vital role in algebra, especially when understanding polynomial equations. Opposite polynomials are two expressions that, when added together, equal zero. Imagine they are like taking a step forward and a step backward – you end up back where you started.

For example, if we have a polynomial like \(y + 7\), its opposite would be \(-y - 7\). When these two are added, \((y + 7) + (-y - 7)\), they cancel each other out resulting in zero. This is similar to the concept of additive inverses in simple arithmetic, where the number 5 and -5 are opposites because their sum is zero.

Understanding opposite polynomials helps in simplifying expressions and solving polynomial equations efficiently.

Addition of Polynomials

Adding polynomials is a straightforward process that involves combining like terms. Each term in a polynomial is made up of a coefficient and a variable that is potentially raised to a power. When adding, you simply align like terms and sum them up, much like standard arithmetic.

Here's a step-by-step guide on adding polynomials:

• Align Like Terms: Start by writing each polynomial in a row, making sure similar terms (like \(x^2\), \(x\), and constants) are positioned above one another.
• Add Coefficients: Sum the coefficients of like terms while retaining the common variable and its power.

For instance, when adding \(2x^2 + 3x + 4\) and \(-x^2 + 2x - 1\), we align:

• \((2x^2 - x^2) + (3x + 2x) + (4 - 1) = x^2 + 5x + 3\)

This makes managing expressions more systematic and easier to handle during calculation.

Zero Sum Condition

The zero sum condition is a simple yet powerful idea that states the sum of opposites results in zero. This is crucial when dealing with polynomial expressions, as it provides a neat way to determine if two expressions negate each other completely.

For polynomials, this means that if you add two polynomials together and arrive at a total of zero, those two polynomials are opposites by definition. This principle underlies the whole process of checking if polynomials are opposites.

• For instance, adding the polynomials \((b-20)\) and \((20-b)\) results in the zero sum \((b - 20 + 20 - b = 0)\). This confirms they are opposites.
• However, if the sum isn't zero, as in adding \(x^2 + 2x - 1\) with \(-x^2 - 2x - 1\) resulting in \(-2\), the polynomials are not considered opposites.

The zero sum condition is essential for simplifying complex polynomial equations and confirming the nature of polynomial relationships.