Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(a^{2} b^{2}-a b+5\right)+\left(a^{2} b^{2}+a b-3\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(2a^{2}b^{2} + 2\).

Step 1: Distribute Signs

The given expression is \((a^{2} b^{2}-a b+5) + (a^{2} b^{2}+a b-3)\). Since there is a plus sign between the two polynomials, you can remove the parentheses without changing the signs of any terms. This simplifies the expression to: \(a^{2} b^{2} - ab + 5 + a^{2} b^{2} + ab - 3\).

Step 2: Combine Like Terms

Add or subtract the coefficients of the like terms. First, for the \(a^{2} b^{2}\) terms: \(a^{2} b^{2} + a^{2} b^{2} = 2a^{2} b^{2}\). Next, for the \(-ab\) and \(+ab\) terms: \(-ab + ab = 0\). Finally, for the constant terms: \(5 - 3 = 2\).

Step 3: Write the Simplified Expression

Using the results from combining like terms, write the final simplified polynomial: \(2a^{2} b^{2} + 2\).

Key Concepts

Combining Like Terms

When dealing with polynomial addition, combining like terms is a crucial step. This means identifying and grouping terms that share the same variables raised to the same power. For example, in the expression \(a^2b^2 - ab + 5 + a^2b^2 + ab - 3\), you need to spot which terms have identical variable parts.

• Both \(a^2b^2\) terms need to be combined because they share the same base and exponent.
• The \(-ab\) and \(+ab\) terms are also like terms, which conveniently add up to zero.
• The numbers \(5\) and \(-3\) are constants, so they also form a pair of like terms.

Once the like terms are correctly identified, you add or subtract their coefficients:- Combine \(a^2b^2\) terms: \(1 + 1 = 2\)- Combine constants: \(5 - 3 = 2\)Doing this correctly sets the stage for the next part of solving polynomial problems.

Simplifying Expressions

Simplifying expressions refers to rewriting them in a form that is shorter or easier to understand without changing their value. Once you have combined like terms, the polynomial expression naturally becomes simpler. Let's look at the expression again:

• After combining like terms, \(a^2b^2 + a^2b^2\) turns into \(2a^2b^2\).
• The \(-ab\) and \(+ab\) terms cancel each other out.
• The simplified form of \(5 - 3\) gives us \(2\).

By putting these together, the expression \(a^2b^2 - ab + 5 + a^2b^2 + ab - 3\) simplifies to \(2a^2b^2 + 2\).
Simplification not only helps in understanding algebraic expressions more clearly, but it also makes further calculations easier.

Distributive Property

The distributive property is a helpful algebraic principle that makes working with expressions more efficient. It involves distributing or spreading out multiplication over addition or subtraction. While this particular exercise doesn't directly utilize the distributive property in an explicit step, understanding it lays a foundation for handling more complex expressions. The principle states that for any three expressions \(a\), \(b\), and \(c\):\[a(b + c) = ab + ac\]This property is often employed when removing parentheses or brackets during operations to ensure every term in the expression is accounted for.

• When dealing with a sum like \((a^2b^2) + (a^2b^2)\), it's as though you have "distributed" the variable \(a^2b^2\) from each term, collecting them together to simplify to \(2a^2b^2\).

Knowing the distributive property well is crucial for algebra since it regularly helps to simplify and solve equations in many math problems.