Question

What is the difference between \(7 a^{2}+3 a b-8 b\) and \(-2 a^{2}+a b-2 b ?\) (A) \(5 a^{2}+4 a b-10 b\) (B) \(9 a^{2}+4 a b-8 b\) (C) \(9 a^{2}+2 a b-6 b\) (D) \(7 a+3 a b-10\)

Short answer

The difference is \(9a^2 + 2ab - 6b\), which matches option (C).

Step-by-step solution

Understand the Problem

We need to find the difference between two expressions: \(7a^2 + 3ab - 8b\) and \(-2a^2 + ab - 2b\). This involves subtracting the second expression from the first.

Set Up the Subtraction

Write the difference expression: \\[(7a^2 + 3ab - 8b) - (-2a^2 + ab - 2b)\]This setup helps us visualize subtracting each term of the second expression from the corresponding term of the first.

Distribute the Negative Sign

Distribute the negative sign across the second expression:\[7a^2 + 3ab - 8b + 2a^2 - ab + 2b\]This changes the sign of each term in the second expression.

Combine Like Terms

Add the like terms from the results:- Combine the \(a^2\) terms: \(7a^2 + 2a^2 = 9a^2\).- Combine the \(ab\) terms: \(3ab - ab = 2ab\).- Combine the \(b\) terms: \(-8b + 2b = -6b\).This gives the simplified expression:\[9a^2 + 2ab - 6b\]

Compare With Choices

Now compare the simplified expression \(9a^2 + 2ab - 6b\) with the given options:- (A) \(5a^2 + 4ab - 10b\)- (B) \(9a^2 + 4ab - 8b\)- (C) \(9a^2 + 2ab - 6b\)- (D) \(7a + 3ab - 10\)The correct choice is (C) because it matches our simplified expression exactly.

Key concepts

Polynomial Subtraction

Polynomial subtraction is a fundamental concept in algebra that involves finding the difference between two polynomial expressions. To perform this operation, we subtract each corresponding term in one polynomial from the other. In the exercise provided, we have two polynomials: \(7a^2 + 3ab - 8b\) and \(-2a^2 + ab - 2b\). Here’s how subtraction works: we take each term from the second polynomial and subtract it from the corresponding term in the first polynomial. However, since the subtraction is applied to the entire expression, we must remember to distribute the negative sign across all terms in the second polynomial. This means that each sign of the terms in the second polynomial is flipped—positives become negatives and negatives become positives. Hence, polynomial subtraction is not just reversing signs; it is about methodically altering the equation to facilitate straightforward subtraction, ensuring each term aligns correctly with its counterpart.

Like Terms

In algebra, like terms are terms that have identical variable parts. This means the variables, together with their respective exponents, must be exactly the same for terms to be considered "like". For instance, in the polynomials from our exercise, the terms with \(a^2\), \(ab\), and \(b\) are groups of like terms.

• The like terms with \(a^2\) are \(7a^2\) and \(2a^2\).
• The like terms involving \(ab\) are \(+3ab\) and \(-ab\).
• Finally, the terms involving \(b\) are \(-8b\) and \(+2b\).

Once the negative sign is distributed among the terms of the second polynomial, these like terms can be added or subtracted directly from each other. Managing like terms simplifies the equation and pares down the problem, making it easier to solve effectively. Identifying and manipulating like terms is a key skill in solving and simplifying algebraic expressions.

Simplifying Expressions

Simplifying expressions is about making algebraic expressions as simple as possible without changing their value. It involves combining like terms after any necessary distribution of negative signs or other operations, like the subtraction in our exercise.After distributing the negative sign in \((7a^2 + 3ab - 8b) - (-2a^2 + ab - 2b)\), we add like terms as follows:

• Combine the \(a^2\) terms: \(7a^2 + 2a^2 = 9a^2\).
• Combine the \(ab\) terms: \(3ab - ab = 2ab\).
• For the \(b\) terms: \(-8b + 2b = -6b\).

The final expression becomes \(9a^2 + 2ab - 6b\), and this is simplified as much as possible. Simplification leads to cleaner, easier-to-read expressions. This is crucial when moving forward with more complex problem-solving, ensuring that you face fewer hurdles by having a streamlined equation to work with.