- Problem 20: Subtraction results in \(2x^2 - 6x\).
- Problem 22: Choice C describes the distributive property.
- Unlabeled Problem: Addition yields \(4x^2 - x + 1\).
- Problem 23: Expansion results in \(x^2 - 4x - 21\).
- Final answers: \(\boxed{2x^2 - 6x}, \boxed{C}, \boxed{4x^2 - x + 1}, \boxed{x^2 - 4x - 21}\).
Explanation
- Problem Analysis and Objectives
We address four distinct algebraic problems involving polynomial subtraction, the identification of algebraic properties, polynomial addition, and binomial expansion. We will solve each one methodically.
- Solving Problem 20: Polynomial Subtraction
To subtract \(5x^2 + 2x\) from \(7x^2 - 4x\), we write the expression as \((7x^2 - 4x) - (5x^2 + 2x)\). First, distribute the negative sign to each term inside the second parentheses: \(7x^2 - 4x - 5x^2 - 2x\). Then, group the like terms: \((7x^2 - 5x^2) + (-4x - 2x) = 2x^2 - 6x\). Thus, the result is \(2x^2 - 6x\).
- Solving Problem 22: Distributive Property Identification
The distributive property states that multiplication distributes over addition: \(a(b + c) = ab + ac\). Let's evaluate the given options:
A\(ab = ba\) represents the commutative property of multiplication.
B\(a(bc) = (ab)c\) represents the associative property of multiplication.
C\(a(b + c) = ab + ac\) is the definition of the distributive property.
D\(a \cdot 1 = a\) represents the identity property of multiplication.
Therefore, option C is the correct answer.
- Solving the Unlabeled Problem: Polynomial Addition
To add \((3x^2 + 4x - 2)\) and \((x^2 - 5x + 3)\), we combine the coefficients of the corresponding powers of \(x\):
- Squared terms: \(3x^2 + x^2 = 4x^2\)
- Linear terms: \(4x - 5x = -x\)
- Constant terms: \(-2 + 3 = 1\)
Combining these, the sum is \(4x^2 - x + 1\).
- Solving Problem 23: Binomial Expansion
To express \((x - 7)(x + 3)\) as a trinomial, we use the FOIL method (First, Outer, Inner, Last):
- First: \(x \cdot x = x^2\)
- Outer: \(x \cdot 3 = 3x\)
- Inner: \(-7 \cdot x = -7x\)
- Last: \(-7 \cdot 3 = -21\)
Summing these results, we get \(x^2 + 3x - 7x - 21\). Simplifying by combining like terms (\(3x - 7x = -4x\)), we obtain \(x^2 - 4x - 21\).
Examples
Algebraic skills like these are fundamental in various fields, such as engineering and economics. For example, when calculating the area of a garden plot whose sides are given by binomial expressions, expanding them into a trinomial allows you to easily plug in values for \(x\) to find the area for different dimensions. Similarly, combining expressions is used to simplify complex cost functions in business modeling, ensuring easier analysis of profit margins.