Problem 37 - College Algebra with Modeling and Visualization

Question

Subtract the polynomials. $$\left(x^{2}-3 x+1\right)-\left(-5 x^{2}+2 x-4\right)$$

Step-by-Step Solution

Answer

The answer is \(6x^2 - 5x + 5\).

Step 1: Distribute the Negative Sign

To subtract one polynomial from another, we must first distribute the negative sign across each term in the second polynomial. This means we have to change the sign of each term in the polynomial \(-5x^2 + 2x - 4\): 1. The negative of \(-5x^2\) is \(5x^2\).2. The negative of \(2x\) is \(-2x\).3. The negative of \(-4\) is \(4\).So, the expression becomes:\(x^2 - 3x + 1 + 5x^2 - 2x + 4\).

Step 2: Combine Like Terms

Now, we combine like terms from both polynomials to simplify the expression:1. Combine the \(x^2\) terms: \(x^2 + 5x^2 = 6x^2\).2. Combine the \(x\) terms: \(-3x - 2x = -5x\).3. Combine the constant terms: \(1 + 4 = 5\).Putting it together, the expression now is:\(6x^2 - 5x + 5\).

Key Concepts

Subtracting Polynomials

Subtracting polynomials might seem tricky at first, but it's quite straightforward once you utilize the correct technique.
When you see two polynomials separated by a minus sign, you need to subtract each term of the second polynomial from the corresponding term in the first polynomial.
This requires attention to the signs and the terms, especially when negative numbers are involved. Remember, subtracting a negative is the same as adding.
For example, given the problem \[x^2 - 3x + 1 - (-5x^2 + 2x - 4),\]the task is to carefully manage the subtraction across the polynomials.

Distributing Negative Sign

One crucial aspect of subtracting polynomials is accurately distributing the negative sign through the second polynomial.
This step is essential because it ensures that each term of the second polynomial is correctly subtracted. Think of it as flipping the sign of every term in the polynomial you're subtracting.

• If the term is negative, it becomes positive.
• If the term is positive, it becomes negative.

In our exercise, we start with the second polynomial: \[-(-5x^2 + 2x - 4)\]Distributing the negative sign, we convert it to:\[5x^2 - 2x + 4\]This step is what turns subtraction into a manageable addition operation.

Combining Like Terms

After managing the signs, the next step is to simplify the expression by combining like terms.
"Like terms" are terms that have the same variables raised to the same power.
When you combine them, you simply add or subtract their coefficients.

• The terms with the same degree must be grouped and combined.
• This way, the polynomial will become more concise.

In this exercise, we have:\[x^2 + 5x^2 = 6x^2\]Combining the linear terms:\[-3x - 2x = -5x\]And for the constant terms:\[1 + 4 = 5\]The result of this process yields a simpler polynomial expression.

Algebraic Simplification

Simplifying an algebraic expression is an important skill in algebra, as it makes the polynomial compact and easy to interpret.
The process involves making sure all like terms are combined, ensuring the expression is in its simplest form.
Through our previous steps, we achieved a consolidated polynomial:\[6x^2 - 5x + 5\]Having followed the steps of distributing negative signs and combining like terms, we arrive at a final, simplified solution.
This form is much easier to work with in further calculations or evaluations.
Achieving clarity and simplicity is the goal, whether for solving equations in algebra or applying these principles in calculus.