Problem 20 - Precalculus Mathematics for Calculus

Question

Find the sum, difference, or product. $$4\left(x^{2}-3 x+5\right)-3\left(x^{2}-2 x+1\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(x^2 - 6x + 17\).

Step 1: Distribute the coefficients

First, apply the distributive property to expand both expressions. For \(4(x^{2}-3x+5)\), distribute 4 to each term in the parenthesis: \(4 \cdot x^2 - 4 \cdot 3x + 4 \cdot 5\). For \(-3(x^{2}-2x+1)\), distribute -3 which gives: \(-3 \cdot x^2 + 3 \cdot 2x - 3 \cdot 1\).

Step 2: Simplify each distributed expression

Simplify the expressions obtained from Step 1. For the first expression, we have \(4x^2 - 12x + 20\). For the second expression after using distribution, we have \(-3x^2 + 6x - 3\).

Step 3: Combine Like Terms

Now, combine like terms from both expanded expressions. Combine the \(x^2\) terms: \(4x^2 - 3x^2 = x^2\). Combine the \(x\) terms: \(-12x + 6x = -6x\). Finally, combine the constant terms: \(20 - 3 = 17\).

Step 4: Write the Final Simplified Expression

After combining like terms, the final simplified expression is \(x^2 - 6x + 17\). This expression is the result of adding and subtracting all terms from the two original expressions.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to multiply a single term across terms inside a parenthesis. It helps in expanding expressions and making complex algebraic operations easier. In Algebra, it is generally used as: \(a(b + c) = ab + ac\).
In our problem, we have two expressions: \(4(x^{2} - 3x + 5)\) and \(-3(x^{2} - 2x + 1)\). For each expression, we use the distributive property to multiply the number outside the parenthesis by each term inside the parenthesis.
For the first expression \(4(x^{2} - 3x + 5)\):

• Multiply 4 by \(x^2\) to get \(4x^2\).
• Multiply 4 by \(-3x\) to get \(-12x\).
• Multiply 4 by 5 to get 20.

The expanded expression becomes \(4x^2 - 12x + 20\).
For the second expression \(-3(x^{2} - 2x + 1)\):

• Multiply -3 by \(x^2\) to get \(-3x^2\).
• Multiply -3 by \(-2x\) to get 6x.
• Multiply -3 by 1 to get -3.

After distribution, the expression is \(-3x^2 + 6x - 3\).
Using the distributive property makes solving algebraic problems step-by-step more manageable and ensures accuracy in calculations.

Combining Like Terms

Combining like terms is an essential process in algebra to simplify expressions. Like terms are terms that have the same variable raised to the same power. When we combine them, we add or subtract their coefficients while keeping the variable part unchanged.
In our exercise, after applying the distributive property, we have two expressions:

• \(4x^2 - 12x + 20\)
• \(-3x^2 + 6x - 3\)

To combine like terms:

• Find the \(x^2\) terms: \(4x^2\) and \(-3x^2\). Combine them to get \((4 - 3)x^2 = x^2\).
• Find the \(x\) terms: \(-12x\) and \(6x\). Combine them to get \((-12 + 6)x = -6x\).
• Combine the constant terms: 20 and \(-3\). 20 - 3 = 17.

This process integrates the important concept of maintaining the expression's integrity while bringing similar parts together. Once mastered, combining like terms is a valuable skill for students as they progress with more advanced algebra topics.

Polynomial Simplification

Polynomial simplification is the process of making a polynomial expression as simple as possible by using algebraic techniques to reduce terms down to their simplest form. Simplification usually involves using both the distributive property and combining like terms.
In our solved exercise, simplification was achieved after expanding the two polynomials and then combining like terms.
The step-by-step simplification process:

• Started by distributing coefficients inside the parenthesis.
• Then combined like terms from the expanded expressions.
• The final result was the simplified expression: \(x^2 - 6x + 17\).

Simplifying polynomials not only makes expressions more manageable but also provides a clearer view of the underlying structure and characteristics of the polynomial. Understanding the simplification process is crucial for solving equations, graphing polynomials, and further algebraic explorations.