Problem 81 - Beginning and Intermediate Algebra

Question

Each of the polynomials below is a polynomial in two variables. Perform the indicated operation(s). $$\begin{aligned} &\left(-12 y^{2} z^{2}+5 y^{2} z-25 y z^{2}+16\right)\\\ &+\left(17 y^{2} z^{2}+2 y^{2} z-15\right) \end{aligned}$$

Step-by-Step Solution

Answer

The short answer is: \(5y^2z^2 + 7y^2z - 25yz^2 + 1\).

Step 1: Identify like terms

First, we need to identify which terms in the polynomials are like terms. Like terms are terms that have the same variables raised to the same powers. The like terms in both polynomials are: - $$-12y^2z^2$$ and $$17y^2z^2$$ - $$5y^2z$$ and $$2y^2z$$ - $$-25yz^2$$ (there is no term like this in the second polynomial) - $$16$$ and $$-15$$ (these are constants) Now that we know which terms are like terms, we can add them together.

Step 2: Add like terms

We add the like terms by adding the coefficients of each pair of like terms together. $$(-12y^2z^2)+(17y^2z^2) = (-12+17)y^2z^2 = 5y^2z^2$$ $$ (5y^2z) + (2y^2z) = (5+2)y^2z = 7y^2z$$ $$ (-25yz^2) = -25yz^2$$ (since there was no like term in the second polynomial) $$(16)+(-15)= (16-15)= 1$$

Step 3: Write the final polynomial

Now that we have added all the like terms, we write the final polynomial by combining all the terms with their new coefficients: $$\begin{aligned} &(5y^2z^2) + (7y^2z) + (-25yz^2) + 1\\ &= 5y^2z^2 + 7y^2z - 25yz^2 + 1 \end{aligned}$$ So, the sum of the given polynomials is: $$5y^2z^2 + 7y^2z - 25yz^2 + 1$$

Key Concepts

Like Terms

Understanding like terms is crucial when working with polynomials. Like terms are simply terms that contain the exact same variables raised to the exact same powers. This means:

• Both the type and the number of variables must be identical.
• The powers of these variables must also be identical.

Examples of like terms include:

• \(3x^2y\) and \(7x^2y\) because both have the same variables \(x\) and \(y\) raised to the same powers.
• \(-9a^3b\) and \(4a^3b\) because both have \(a^3\) and \(b\) as common attributes.

Recognizing like terms allows you to combine them by adding or subtracting their numerical coefficients. When performing operations with polynomials, like terms are the key to simplifying complex expressions efficiently.

Adding Polynomials

Adding polynomials involves combining like terms to produce a new polynomial. The process is straightforward:

• List and identify all terms as you scan through the polynomials.
• Group terms with identical variable parts — these are your like terms.
• Add or subtract the coefficients of these like terms.

For instance, if you encounter the expression:\[ (3x^2 + 2x + 5) + (5x^2 + 3x - 1) \]

• Identify the like terms: \(3x^2\) and \(5x^2\); \(2x\) and \(3x\); and \(5\) and \(-1\).
• Combine them as follows: \((3+5)x^2, (2+3)x, (5-1)\).

The final simplified expression is \(8x^2 + 5x + 4\).This method showcases the importance of correctly identifying and working with like terms to simplify and find the result of polynomial expressions.

Two-Variable Polynomials

Polynomials with two variables, such as in the original problem, introduce additional complexity compared to single-variable polynomials. These polynomials include terms with combinations of two different variables, which can be raised to various powers.

• A standard example might look like \(-12y^2z^2 + 5y^2z - 25yz^2 + 16\).
• Each term can share parts of the variables or differ entirely.

When working with two-variable polynomials:

• Ensure you only combine like terms, which have the same set of variables raised to the same powers, as shown in the worked example.
• Pay close attention to coefficients as you add terms across separate polynomials.

Handling two variables requires you to be detailed and careful about variable combinations. However, the satisfaction of arriving at simplified terms post-operation can be quite rewarding.