Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Subtract \(7 x^{3}-4 x+2\) from \(4 x^{2}-5 x+10\)

Short answer

-7x^3 + 4x^2 - x + 8

Step-by-step solution

Write down the expressions

First, write down the two polynomial expressions: the one being subtracted (\(7x^3 - 4x + 2\)) and the one it is being subtracted from (\(4x^2 - 5x + 10\)).

Set up the subtraction

Arrange the polynomials in a way that aligns like terms vertically. Write the polynomial \(4x^2 - 5x + 10\) first, then subtract \(7x^3 - 4x + 2\) from it, ensuring to distribute the subtraction sign correctly.

Distribute the subtraction

Apply the negative sign to each term in the polynomial being subtracted: \[- (7x^3 - 4x + 2) = -7x^3 + 4x - 2\].

Combine like terms

Now, combine the like terms from both polynomials: degree 3: \(0 - 7x^3 = -7x^3\)degree 2: \(4x^2 + 0 = 4x^2\)degree 1: \(-5x + 4x = -x\)constant term: \(10 - 2 = 8\)

Write the result

The final result after subtraction is obtained by combining all the simplified terms: \(-7x^3 + 4x^2 - x + 8\).

Key concepts

Polynomial Operations

Polynomials are algebraic expressions that consist of variables and coefficients arranged in terms of powers. Common operations with polynomials include addition, subtraction, multiplication, and division. Each term in a polynomial is a product of a coefficient and a variable raised to a non-negative integer power. For example, in the polynomial equation \(4x^2 - 5x + 10\), the terms are \(4x^2\), \(-5x\), and \(10\). When it comes to polynomial operations, especially subtraction, it's crucial to align like terms and correctly distribute any minus signs. This helps in ensuring accuracy and avoiding mistakes. Always pay close attention to the signs (+/-) in each term while performing operations. Ensure each step is carefully double-checked to keep your work error-free.

Combining Like Terms

Combining like terms is a method used to simplify polynomials by adding or subtracting terms with the same variables raised to the same power. For instance, when handling the exercise, we grouped terms like \(-5x\) and \(4x\) because they both have the variable \(x\) to the first power. The same goes for the constants and other higher-degree terms. Always arrange terms in descending order of their degrees for clarity. Combining like terms greatly simplifies polynomial expressions and makes it easier to work with them in subsequent steps. Important tips to remember include:

• Like terms have exactly the same variables and exponents.
• Simplify by operating only on the coefficients.
• Maintain the correct sign before each term to avoid mistakes.

Applying these rules makes working with complex equations more manageable and ensures you get to the correct, simplified form of the polynomial.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (addition, subtraction, multiplication, and division). Unlike simple numbers, they use variables to represent values that can change. In the context of polynomial subtraction, these expressions involve multiple terms each with a variable raised to a certain power. For example, in the problem \(7x^3 - 4x + 2\), each term represents a part of the algebraic expression. Understanding algebraic expressions requires familiarity with the fundamental concepts of algebra:

• Variables: Represent unknown values and can change.
• Constants: Fixed values that do not change.
• Coefficients: Numbers that multiply the variables.

Mastering how to manipulate algebraic expressions is key in solving more complex algebra problems and is a foundational skill in mathematics. With practice, recognizing patterns and common techniques, like distribution and combining like terms, will become second nature.