Problem 52 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(-c^{5}+5 c^{4}-12\right)-\left(2 c^{5}-c^{4}\right) $$

Step-by-Step Solution

Answer

The result is \(-3c^{5} + 6c^{4} - 12\).

Step 1: Remove Parentheses

To subtract one polynomial from another, we first remove the parentheses. Remember to distribute the negative sign before the second set of parentheses across each term inside that set:\[(-c^{5} + 5c^{4} - 12) - (2c^{5} - c^{4}) = -c^{5} + 5c^{4} - 12 - 2c^{5} + c^{4}\]

Step 2: Combine Like Terms

Next, we combine like terms by adding or subtracting the coefficients of terms with the same power of \(c\):- Combine \(-c^{5}\) and \(-2c^{5}\): \(-c^{5} - 2c^{5} = -3c^{5}\)- Combine \(5c^{4}\) and \(c^{4}\): \(5c^{4} + c^{4} = 6c^{4}\)- The constant term \(-12\) remains unchanged as there are no other constant terms.Now, we have the result:\[-3c^{5} + 6c^{4} - 12\]

Key Concepts

Distribute Negative Sign

Distributing the negative sign is a crucial step when you're subtracting polynomials. It ensures each term in the polynomial is correctly subtracted. Take the expression \((2c^{5} - c^{4})\): when the expression is preceded by a negative sign, each term inside must be reversed in sign.

So,

• \(2c^{5}\) becomes \(-2c^{5}\)
• \(-c^{4}\) becomes \(+c^{4}\)

This action essentially "flips" the polynomial over, changing all positive terms negative and vice versa. Think of it as multiplying the entire expression by \(-1\).
Once the sign is distributed, the subtraction problem transforms into a simple addition of terms with their new signs. This step ensures no mistakes occur in reducing the expression further.

Combine Like Terms

Once any needed signs are distributed, the next step in polynomial subtraction is to combine like terms. These are terms with the same variables raised to the same powers. Only these like terms can be combined.

In our calculation:

• For the terms in \(-c^{5}\) and \(-2c^{5}\), you combine them: \(-c^{5} - 2c^{5} = -3c^{5}\).
• For \(5c^{4}\) and \(c^{4}\): \(5c^{4} + c^{4} = 6c^{4}\).
• The constant \(-12\) stands alone as there are no like terms to combine with it.

Combining like terms helps simplify the expression significantly, making it easier to work with or interpret in future calculations.

Polynomial Expression

A polynomial expression is an algebraic expression consisting of variables, coefficients, and operations involving addition, subtraction, multiplication, and non-negative integer exponents of variables. In our example, \(-3c^{5} + 6c^{4} - 12\), we have a polynomial with terms involving the variable \(c\) raised to various powers.

Understanding polynomials involves:

• Recognizing each term is made of a coefficient and variables.
• The coefficients (like \(-3\) and \(6\)) are numbers that multiply the variables.
• Each term's degree is decided by the exponent of the variable, and the term with the highest degree (here \(-3c^{5}\)) often dictates the polynomial's degree.

Mastering polynomials is foundational in algebra, as they appear in many different mathematical contexts, simplifying complex problems and equations into manageable terms.