Problem 38 - Algebra 1: Concepts and Skills

Question

Use a vertical format to add or subtract. $$ \left(4 x^{2}-7 x+2\right)+\left(-x^{2}+x-2\right) $$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(3x^{2} - 6x\).

Step 1: Identification of Like Terms

Like terms in an expression are terms with exactly the same variable factors. So in this case, \(4x^{2}\) and \(-x^{2}\) are like terms, \(-7x\) and \(x\) are like terms, and \(2\) and \(-2\) are like terms.

Step 2: Addition/Subtraction of Like Terms

Once like terms are identified, the coefficients of those terms are added or subtracted based on the operation between them. Here, we add or subtract the coefficients of the like terms identified in Step 1: \( (4x^{2} - x^{2}),\) \((-7x + x)\), and \( (2 - 2) \).

Step 3: Simplification

When the addition or subtraction of like terms is done, simplify the expression. In our case, simplifying yields: \( 3x^{2} - 6x + 0\) or \( 3x^{2} - 6x\).

Key Concepts

Like Terms

When dealing with polynomials, understanding like terms is crucial for simplifying expressions. Like terms have the same variables raised to the same power. For instance, in the expression \(4x^{2} - 7x + 2\), both \(4x^{2}\) and \(-x^{2}\) are like terms because they each have the variable \(x\) squared. Similarly, \(-7x\) and \(x\) are like terms because they both have the variable \(x\) raised to the first power. Furthermore, constant terms like \(2\) and \(-2\), which have no variable part, are also considered like terms.

In general, when identifying like terms in any polynomial expression, focus on these characteristics:

• The variable component must match exactly.
• The exponent of the variable must be the same for the terms.
• Only the coefficients in front of the terms are added or subtracted.

This understanding enables us to combine them effectively through addition or subtraction.

Addition of Polynomials

The addition of polynomials is straightforward when like terms have been identified. This process involves adding up the coefficients of the like terms while keeping the variable part unchanged. Let's take the problem: \( (4x^{2}-7x+2)+(-x^{2}+x-2) \). First, align the polynomials so that like terms are positioned in the same column:

• \(4x^{2}\)
• \(-7x\)
• \(+2\)

and

• \(-x^{2}\)
• \(+x\)
• \(-2\)

When you add these polynomials:

• For \(x^{2}\) terms: \(4x^{2} + (-x^{2}) = 3x^{2}\).
• For \(x\) terms: \(-7x + x = -6x\).
• For constants: \(2 + (-2) = 0\).

Thus, the addition of the given polynomials yields \(3x^{2} - 6x\). Notice that constant terms may cancel each other out, as seen here.

Subtraction of Polynomials

Subtracting polynomials can be seen as adding the opposite. This means that when subtracting, you distribute a minus sign (effectively changing the signs) across the terms of the polynomial being subtracted. Consider the expressions: \(4x^{2} - 7x + 2\) and \(-x^{2} + x - 2\). If you want to subtract, you change the signs of the second polynomial:

• \(-x^{2}\) becomes \(+x^{2}\)
• \(+x\) becomes \(-x\)
• \(-2\) becomes \(+2\)

This converts the operation to addition with the inverted polynomial:

• \(4x^{2} - 7x + 2 + x^{2} - x + 2\)

Now you can simply add them just as explained in the addition of polynomials:

• Combine \(x^{2}\) terms: \(4x^{2} + x^{2} = 5x^{2}\).
• Combine \(x\) terms: \(-7x - x = -8x\).
• Combine constant terms: \(2 + 2 = 4\).

Resulting in \(5x^{2} - 8x + 4\). Thus, subtracting polynomials becomes another case of combining like terms after sign adjustment.