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Chapter 4: Problem 111

Simplify. $$-\left(6 a^{2}-7 b^{2}\right)$$

Short Answer

Expert verified
The simplified form of the expression is \(-6a^{2} +7b^{2}\).

Step by step solution

01

Distribute

Lets start with applying the distributive property - multiplying each term inside the parentheses by the negative sign. So it becomes: \(-6a^{2} +7b^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
Applying the distributive property can be a bit tricky when dealing with negative coefficients. It's important to remember that multiplying by a negative number changes the sign of the term being multiplied. So if you have \( -x(y - z) \), it's equivalent to \( -xy + xz \) because you're distributing \( -x \) across \( y \) and \( -z \) giving you the subtraction of \( xy \) and the addition of \( xz \) once the sign of \( -z \) shifts.
Negative Signs in Algebra
When you see a subtraction sign in front of the parentheses, like in our original exercise, apply the negative sign to each term within the parentheses individually. It's a common mistake to only apply it to the first term or to forget it completely. Always keep in mind that every term gets affected. This makes dealing with negatives straightforward, and once you make it a habit, it becomes second nature.
Polynomials
Simplifying polynomials requires applying the distributive property correctly and understanding how to handle signs. The terms in polynomials can be rearranged by addition or subtraction due to their commutative property. However, if you're multiplying a polynomial by a number (or another polynomial), ensure each term is multiplied. Remember that understanding the structure of polynomials and how they can be simplified is essential to mastering algebra.

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