Problem 47 - Algebra 1: Concepts and Skills

Question

Use a vertical format or a horizontal format to add or subtract. $$ \left(3 x+2 x^{2}-4\right)-\left(x^{2}+x-6\right) $$

Step-by-Step Solution

Answer

The resultant polynomial after subtraction is \( x^{2} + 2x + 2 \)

Step 1: Distribute the negative sign

Firstly, distribute the negative sign across all the terms of the subtracted polynomial to switch the signs. \[ (3 x+2 x^{2}-4) - (x^{2}+x-6) \] becomes \[ 3 x +2 x^{2} - 4 - x^{2} - x + 6 \]

Step 2: Combine like terms

Then, look for like terms (terms having the same term variable and degree) and combine them. The like terms here are \(2x^{2}\) and \(- x^{2}\) and \(3x\) and \(- x\). Combine those to get \[ x^{2} + 2x + 2 \]

Step 3: Write final result

Finally, rewrite the solution in descending powers of x which is the common way to write polynomial expressions. The final answer will be \[ x^{2} + 2x + 2 \]

Key Concepts

Combining Like Terms

When it comes to algebra, combining like terms is a fundamental skill that simplifies expressions and equations. Like terms are terms that have the same variable raised to the same power. In other words, their variables and their exponents are identical. For example, the terms \(2x^{2}\) and \(5x^{2}\) are like terms because they both contain the variable \(x\) raised to the second power. However, \(2x^{2}\) and \(3x\) are not like terms since the exponents differ.

To combine like terms, simply add or subtract the coefficients (the numerical parts) of these terms while keeping the variable part unchanged. Suppose you have the expression \(7x + 3 - 2x + 5\), the like terms \(7x\) and \(2x\) can be combined by subtracting their coefficients: \(7 - 2 = 5\), resulting in \(5x\). The final simplified form of the expression would be \(5x + 8\), with \(3\) and \(5\) being combined as constants. Understanding how to merge like terms efficiently is essential for successful polynomial subtraction.

Subtraction of Polynomials

Subtracting polynomials is an operation that falls under the broader umbrella of polynomial arithmetic. To properly subtract one polynomial from another, align them by their respective like terms—either using a vertical or horizontal method—and then subtract corresponding coefficients. For instance, in the expression \(\left(3 x+2 x^{2}-4\right)-\left(x^{2}+x-6\right)\), it's crucial to mind the order of subtraction to ensure accuracy.

Here's a useful tip: when stacking polynomials vertically for subtraction, be extremely careful with the signs. It's a common mistake to overlook the change of sign when distributing the subtraction over the second polynomial. To avoid errors, some prefer to rewrite the subtraction problem as an addition problem by distributing a negative one (\(-1\)) to every term in the second polynomial, then proceed to combine like terms.

Distributive Property

The distributive property is a key concept in algebra that allows for the multiplication of a single term by each term within a parenthesis. It's represented by the formula \(a(b + c) = ab + ac\). This property is particularly useful when dealing with the subtraction of polynomials, as seen with the term \-1\ that's distributed across the subtracted polynomial.

Consider the expression from our earlier example, \(\left(3 x+2 x^{2}-4\right)-\left(x^{2}+x-6\right)\), where the distributive property is applied to eliminate the parentheses. By distributing the negative sign across each term in the second polynomial, we effectively change the signs of its terms, transforming our expression into a simpler addition problem. With the negative sign properly distributed as \(\left(-x^{2}-x+6\right)\), it’s much easier to proceed with combining like terms without any oversight. This small step is critical for avoiding common errors in polynomial subtraction.