Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(3+2 b+b^{2}\right)-\left(9+5 b+b^{2}\right) $$

Step-by-Step Solution

Answer

The simplified polynomial is \(-6 - 3b\).

Step 1: Distribute the Negative Sign

Apply the distributive property to remove the parentheses by distributing the negative sign through each term of the second polynomial. This means we take each term in \(9 + 5b + b^2\) and change its sign. Thus, it becomes \(-9 - 5b - b^2\).

Step 2: Combine the Polynomials

Now write out the entire expression after distributing the negative signs: \((3 + 2b + b^2) - (9 + 5b + b^2) = 3 + 2b + b^2 - 9 - 5b - b^2\).

Step 3: Group Like Terms

Group the terms by their degrees (constant terms, linear terms, and quadratic terms) before combining them: \((3 - 9) + (2b - 5b) + (b^2 - b^2)\).

Step 4: Simplify Each Group

Calculate each grouped expression:- For the constant terms: \(3 - 9 = -6\).- For the linear terms: \(2b - 5b = -3b\).- For the quadratic terms: \(b^2 - b^2 = 0\).

Step 5: Write the Simplified Polynomial

Combine the simplified terms to write the polynomial in its simplest form: The final expression is \(-6 - 3b\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps you manage expressions with parentheses. When subtracting polynomials, like in the exercise given, it's essential to distribute any negative signs that precede a polynomial. This ensures that each term inside the parentheses is correctly adjusted.
In the case of our problem, you begin with:

• The expression inside the parentheses: \((9 + 5b + b^2)\).
• When you distribute the negative sign across this polynomial, each term inside changes its sign.
• So, it becomes: \(-9 - 5b - b^2\).

Once the signs are distributed correctly, you can proceed with polynomial subtraction. This prevents mistakes and simplifies future calculations.

Combining Like Terms

Once you've dealt with distributing signs, the next crucial step is to combine like terms. Like terms refer to terms that have the same variables raised to the same power. Without combining them, you can't simplify the expression effectively.
In our problem, we regroup the expression as:

• Constants: Combine \(3\) and \(-9\).
• Linear terms: Combine \(2b\) and \(-5b\).
• Quadratic terms: Combine \(b^2\) and \(-b^2\).

Grouping these terms ensures that all similar components are together, making the mathematical operations straightforward. This allows for an easier transition to simplifying the expression.

Simplifying Expressions

Simplifying expressions is the final step in cleaning up your polynomial. It involves performing actual arithmetic on each group of like terms. This clarity helps you see the final form of the expression, free from superfluous terms.
For the example given:

• Constants: Calculate \(3 - 9 = -6\).
• Linear terms: Calculate \(2b - 5b = -3b\).
• Quadratic terms: Calculate \(b^2 - b^2 = 0\).

Combining these results, the simplified polynomial is \(-6 - 3b\). Each term has been reduced so that no further simplification is possible, giving a clear and concise expression. By understanding how to simplify expressions, you can ensure that your mathematical solutions are neat and easily interpretable.