Problem 74 - Algebra for College Students

Question

Explain how to subtract the polynomial \(-3 x^{2}+2 x-4\) from \(4 x^{2}+6 .\)

Step-by-Step Solution

Answer

The result is \(7x^2 - 2x + 10\).

Step 1: Write Down the Polynomials

First, clearly write down the polynomials involved in the subtraction. We have to subtract \(-3x^2 + 2x - 4\) from the polynomial \(4x^2 + 6\). This can be formulated as: \[(4x^2 + 6) - (-3x^2 + 2x - 4).\]

Step 2: Distribute the Negative Sign

Distribute the negative sign across the second polynomial. This means changing the sign of each term in the polynomial being subtracted: \[-(-3x^2) = +3x^2\], \[-2x = -2x\], and \[-(-4) = +4\]. Thus the expression becomes:\[4x^2 + 6 + 3x^2 - 2x + 4.\]

Step 3: Combine Like Terms

Now, combine the like terms:- Combine the \(x^2\) terms: \(4x^2 + 3x^2 = 7x^2\).- Since there is only one \(x\) term, it remains as \(-2x\).- Combine the constant terms: \(6 + 4 = 10\).Thus, the simplified expression is:\[7x^2 - 2x + 10.\]

Step 4: Write the Result

Finally, write down the result of the subtraction: \(7x^2 - 2x + 10\). This is the polynomial after subtracting \(-3x^2 + 2x - 4\) from \(4x^2 + 6\).

Key Concepts

Combining Like Terms

When working with polynomials, combining like terms is a crucial step for simplifying expressions. Like terms are those terms that have exactly the same variable component with the same exponent, like

• \(x^2\) terms can only be combined with other \(x^2\) terms.
• \(x\) terms get combined with \(x\) terms.
• Constant numbers or terms without variables are combined with each other.

This makes polynomials easier to handle and further simplifies calculations. In the problem provided, the terms \(4x^2\) and \(3x^2\) are like terms because they both involve \(x^2\). By combining them, we get \(7x^2\). The constant terms, \(6\) and \(4\), also combine to give \(10\). However, since there is only one \(x\) term, it remains

• -2x

as is. The result is the simplified expression: \(7x^2 - 2x + 10\). This demonstrates the power and utility of combining like terms, making the expression shorter and easier to work with.

Distributing Negative Signs

Distributing negative signs is an essential mathematical operation, especially when subtracting polynomials. A negative sign in front of a polynomial needs to be distributed across all terms within the polynomial. This process involves changing the sign of each term:

• If a term is positive, it becomes negative.
• If it is negative, it turns positive.

For example, given the polynomial expression \[- (-3x^2 + 2x - 4)\], applying the negative sign leads to:

• \(-(-3x^2)\) becomes \(+3x^2\).
• \(-2x\) stays as \(-2x\).
• \(-(-4)\) becomes \(+4\).

Thus, the process results in changing \[4x^2 + 6 - (-3x^2 + 2x - 4)\] to \[4x^2 + 6 + 3x^2 - 2x + 4.\] Effectively managing the distribution of negative signs ensures accurate and simplified results in polynomial operations.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra that makes complex problems manageable. It involves both combining like terms and distributing negative signs effectively. By simplifying, you reduce an expression to its most straightforward form, making calculations quicker and interpretation clearer. Consider the expression \[4x^2 + 6 + 3x^2 - 2x + 4.\]

Step-by-Step Simplification:

• Combine the \(x^2\) terms: \(4x^2 + 3x^2 = 7x^2\).
• The linear term remains \(-2x\) as there are no other like terms for it.
• Add the constant terms: \(6 + 4 = 10\).

The resulting simplified expression is \(7x^2 - 2x + 10\). By focusing on these simplifying strategies, complex equations can be broken down into simpler parts. This approach aids in solving a wide range of algebraic problems efficiently by minimizing the amount of work.