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Chapter 10: Problem 628

Simplify each expression. $$\left(10 x^{2}-3 x+5\right)-\left(4 x^{2}-6\right)$$

Short Answer

Expert verified
6x^2 - 3x + 11

Step by step solution

01

Distribute the Negative Sign

Distribute the negative sign to each term inside the parentheses. This changes the sign of each term in the second expression: \ \[ \begin{align*} (10x^2 - 3x + 5) - (4x^2 - 6) & = 10x^2 - 3x + 5 - 4x^2 + 6 \ \end{align*} \]
02

Combine Like Terms

Combine the terms that have the same variables and exponents. Add the coefficients of the like terms: \ \[ \begin{align*} 10x^2 - 4x^2 & = 6x^2 \ -3x & = -3x \ 5 + 6 & = 11 \ \text{So, the simplified expression is:} & \ 6x^2 - 3x + 11 \ \end{align*} \]

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Key Concepts

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

Combining Like Terms
Combining like terms is a key concept in simplifying algebraic expressions. Like terms are terms that have the same variable and the same exponent. For example, in the expression 10x^2 - 4x^2 - 3x + 5,x^2 terms (10x^2, 4x^2), and constant terms (5, 6) are like terms.
Combining like terms means adding or subtracting the coefficients (the numerical parts) of these terms.
This helps to reduce the expression to a simpler form.
Let's take the first step from the solution.
    - 10x^2 and - 4x^2 combine to give 6x^2- The term -3x remains the same because there is no other x term- 5 and 6 combine to give 11
Combining like terms makes algebraic expressions easier to handle.
Distributing Negative Signs
Distributing negative signs can often be a tricky part of algebra. When you see a negative sign in front of a parenthesis, it means you need to multiply each term inside the parenthesis by -1.
For example, in the given solution before distributing the negative sign, we have: (10x^2 - 3x + 5) - (4x^2 - 6).
Distributing the negative sign inside the second set of parentheses, we convert - (4x^2 - 6) into: -4x^2 + 6.

This means we now have: 10x^2 - 3x + 5 - 4x^2 + 6.
By correctly distributing the negative sign, we ensure that all subsequent steps of simplification are accurate.
Polynomial Simplification
Polynomial simplification involves using both combining like terms and distributing negative signs. By following these steps accurately, you can simplify any polynomial expression to its simplest form.
In our example, after distributing the negative sign and combining like terms, we are left with a simpler polynomial expression: 10x^2 - 3x + 5 - 4x^2 + 6.
Combining like terms from here, we perform the following:
    - 10x^2 - 4x^2 = 6x^2- -3x remains the same- 5 + 6 = 11
This results in the simplified polynomial: 6x^2 - 3x + 11.
Polynomial simplification makes the expressions more manageable and prepares them for further algebraic operations or problem-solving.

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