Question

Perform the indicated operation. $$ \left(2 x^5+x^4-4 x^2\right)-\left(x^5-3\right) $$

Short answer

The result of the given operation is \(x^5 + x^4 - 4x^2 + 3\).

Step-by-step solution

Identify corresponding terms

In the given expression, the terms that correspond to each other based on their power are: \(2x^5\) and \(x^5\) (with degree 5), \(x^4\) (with degree 4), \(-4x^2\) and \(-3\) (constant term, i.e., degree 0).

Perform the subtraction

Now, subtract the like terms. Subtract \(x^5\) from \(2x^5\) to get \(x^5\). Subtract the constants. None is present in the first polynomial, so it's like subtracting \(-3\) from \(0\), yielding \(3\). The term \(x^4\) and \(-4x^2\) from the first polynomial do not have corresponding terms in the second polynomial, so they should be kept as it is.

Write the final answer

Now, gather all the results together to form the final polynomial after subtraction. The final polynomial becomes \(x^5 + x^4 - 4x^2 + 3\).

Key concepts

Polynomial Subtraction

Subtracting polynomials is a fundamental operation in algebra, where you subtract one polynomial from another. This involves distributing a negative sign and combining like terms.

In the given exercise, we need to subtract one polynomial from another:

• Start by writing the entire second polynomial with a negative sign in front of it.
• Once this is done, look at the terms that can be subtracted, such as those with the same power (like terms).
• After applying the negative sign, use addition of opposite signs to simplify.

This mastery of polynomial subtraction is vital in mathematics, as it prepares you for solving more complex equations and helps in understanding polynomial functions thoroughly.

Like Terms

Like terms in polynomials are terms that contain the same variable raised to the same power. It's essential to only combine terms that are truly 'like' to simplify expressions accurately.

In our example, identify the terms in both polynomials:

• Both polynomials have terms that include powers of 5, which are like terms.
• From the first polynomial, the term with power of 4 stands alone, as does the term with power of 2.
• Numbers with no variables (constants) are also considered like terms, even if they are from separate polynomials.

Recognizing like terms correctly makes subtraction much simpler and prevents common errors in combining unrelated terms.

Polynomial Degree

A polynomial's degree is the highest power of the variable in the expression. Recognizing the degree is important for various algebraic operations and solutions.

For the polynomials given, the degree is determined by the highest exponent:

• The term with the highest exponent is often the most significant in determining the behavior of the polynomial.
• In our example, the highest degree is 5, due to the term with \(x^5\).
• While dealing with subtraction, always line up the terms in descending order of degree before performing the operation.

Understanding degrees ensures you can effectively categorize and work with polynomials, leading to successful simplifications and solutions.

Mathematics Education

Learning polynomial operations is a crucial aspect of mathematics education, facilitating skill-building in algebra.

Students encounter various polynomials and must learn operations such as addition, subtraction, and multiplication to succeed in mathematics and related fields.

• These operations form the basis for understanding more complex mathematical concepts.
• Education focuses on teaching problem-solving methods, accuracy, and simplification of expressions.
• Regular practice with polynomial operations helps students understand abstract algebraic concepts better and prepares them for advanced courses.

Mathematics education aims to develop critical thinking and analytical skills, which are not only useful in math but in real-world problem-solving as well.