Problem 60 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left.7 a^{2}-3 b^{2}+10\right)-\left(-2 a^{2}+b^{2}-12\right) $$

Step-by-Step Solution

Answer

The result is \(9a^2 - 4b^2 + 22\).

Step 1: Distribute Negative Sign

Distribute the negative sign across the second expression. This changes the signs of all terms inside the second set of parentheses:\(-(-2a^2 + b^2 - 12)\) becomes \(2a^2 - b^2 + 12\).

Step 2: Combine Like Terms

Now add the modified second expression \(2a^2 - b^2 + 12\) to the first expression \(7a^2 - 3b^2 + 10\):- Combine the \(a^2\) terms: \(7a^2 + 2a^2 = 9a^2\).- Combine the \(b^2\) terms: \(-3b^2 - b^2 = -4b^2\).- Combine the constant terms: \(10 + 12 = 22\).

Step 3: Write the Final Expression

Combine all the results from the previous step to form the final expression:\(9a^2 - 4b^2 + 22\).

Key Concepts

Polynomials

In algebra, a polynomial is a type of algebraic expression that involves sums and/or differences of terms. These terms usually consist of variables raised to whole-number exponents, coefficients (numbers multiplying these variables), and sometimes constants. Polynomials have varying degrees, which are defined by the highest exponent of the variable in the expression. For example, in the exercise given:

• The term \(7a^2\) includes the variable \(a\) raised to the power of 2, with 7 as the coefficient.
• Similarly, \(-3b^2\) has the variable \(b\) squared, and the coefficient is -3.
• The constant term is 10, a number without a variable.

The beauty of polynomials lies in their structured arrangement and their versatility in algebraic operations like addition, subtraction, and multiplication. Mastery of polynomials is crucial for progressing in algebra and beyond.

Like Terms

Like terms in algebra are terms within an algebraic expression that have the same variable raised to the same power. Only the coefficients—the numbers in front of these variables—might be different. Recognizing and combining like terms is a fundamental skill needed in simplifying expressions and solving equations.

• For example, the terms \(7a^2\) and \(2a^2\) are like terms because they both contain \(a^2\).
• In contrast, \(b^2\) terms like \(-3b^2\) and \(-b^2\) can be combined as they both involve \(b^2\).

The process of adding or subtracting like terms is straightforward: you keep the common variable part unchanged and perform the arithmetic operation on the coefficients. Here, you added the \(a^2\) terms to get \(9a^2\) and the \(b^2\) terms to get \(-4b^2\). Remember, it's essential that terms be "like"; otherwise, they cannot be combined directly.

Distributive Property

The distributive property is a useful algebraic rule for expanding and simplifying expressions. It states that multiplying a sum by a number is equivalent to multiplying each addend by the number and then adding or subtracting the results.

• For example, if you have \(a(b + c)\), it distributes to \(ab + ac\).
• In subtraction, like in the given exercise, it means reversing the signs in the expression within parentheses when preceded by a negative sign: \(-(-2a^2 + b^2 - 12)\) becomes \(2a^2 - b^2 + 12\).

The distributive property is particularly helpful for eliminating parentheses and simplifying expressions for further calculations. It's a foundational principle in algebra because it allows the manipulation of expressions into usable forms for solving more complex algebraic equations.