Problem 46 - Algebra for College Students

Question

Perform the operations as described. Subtract the sum of \(-6 n^{2}+2 n-4\) and \(4 n^{2}-2 n+4\) from \(-n^{2}-n+1\).

Step-by-Step Solution

Answer

The result is \(n^2 - n + 1\).

Step 1: Find the Sum

First, we need to find the sum of the two expressions \(-6n^2 + 2n - 4\) and \(4n^2 - 2n + 4\). By adding these expressions term by term:\[(-6n^2 + 4n^2) + (2n - 2n) + (-4 + 4)\]This simplifies to: \[-2n^2 + 0n + 0\] which is simply \(-2n^2\).

Step 2: Subtract the Sum from the Given Expression

Now, we subtract the sum obtained in the first step from the expression \(-n^2 - n + 1\).To do this subtraction, change the sign of \(-2n^2\):\[-(-2n^2) = +2n^2\].Then perform the subtraction:\[-n^2 - n + 1 + 2n^2\]Combine like terms:\[(2n^2 - n^2) - n + 1\]This gives us:\[n^2 - n + 1\].

Key Concepts

Subtraction of Polynomials

Subtraction of polynomials is just like subtracting numbers, but with terms that include variables. You have to pay attention to the signs. When subtracting polynomials, first change the signs of the terms in the expression to be subtracted. Then, add the results to the first polynomial.

In the given exercise, we had to subtract \(-2n^2\) (after adding two expressions) from \(-n^2 - n + 1\). To do this, we changed the sign of \(-2n^2\) to \(+2n^2\), and then combined it with the original expression.

• Change subtraction to addition by altering signs
• Combine the two expressions

Remember, always perform operations on like terms to achieve the correct result.

Addition of Polynomials

Adding polynomials is a crucial skill. It involves simply combining like terms from each polynomial. This means that terms with the same power or exponent are added together.

In the task, we first needed to find the sum of the given polynomials \(-6n^2 + 2n - 4\) and \(4n^2 - 2n + 4\) by adding the terms of each with the corresponding terms from the other polynomial.

• Add coefficients of the same degree terms
• Write out the new polynomial after summing

The result of \(-6n^2 + 4n^2\), \(2n - 2n\), and \(-4 + 4\) was \(-2n^2\). Always align terms with the same power when performing such operations.

Combining Like Terms

Combining like terms is an essential part of simplifying polynomials. "Like terms" are those terms that have the exact same variables raised to the same power.

In the solution, after subtracting the polynomials, we were left with \(-n^2 + 2n^2 - n + 1\). Here, we needed to identify and combine the like terms to simplify the expression.

• Identify terms with the same variable and exponent
• Add or subtract their coefficients

As in our exercise, the terms \(2n^2 - n^2\) were combined to give \(n^2\). It's always important to break down expressions to their simplest form by combining all similar parts.