Problem 20 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ 4\left(x^{2}-3 x+5\right)-3\left(x^{2}-2 x+1\right) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Constants

Distribute the constants 4 and -3 across the terms within each parenthesis. This means multiplying 4 with each term inside the first parenthesis and -3 with each in the second parenthesis: \[4(x^2 - 3x + 5) = 4x^2 - 12x + 20\]\[-3(x^2 - 2x + 1) = -3x^2 + 6x - 3\]

Step 2: Combine Like Terms

Now, add the resulting expressions from Step 1 together by combining like terms. Like terms are terms that have the same variable raised to the same power. Combine the expressions:\[4x^2 - 12x + 20 - 3x^2 + 6x - 3\]Combine like terms:- Combine the \(x^2\) terms: \(4x^2 - 3x^2 = x^2\)- Combine the \(x\) terms: \(-12x + 6x = -6x\)- Combine the constant terms: \(20 - 3 = 17\)

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows us to simplify expressions and solve equations by multiplying terms correctly. In essence, it means multiplying a single term by all terms inside a parenthesis. This property can be written as:

• For any numbers or expressions, if you have: \( a(b + c) = ab + ac \)
• It means you distribute the term \( a \) across both \( b \) and \( c \), resulting in two separate products.

Let’s look at our exercise as an example to understand this concept better:
When we apply the Distributive Property to \( 4(x^2 - 3x + 5) \), we distribute 4 to each term inside the parenthesis:

• Multiply 4 by \( x^2 \), getting \( 4x^2 \).
• Multiply 4 by \( -3x \), resulting in \( -12x \).
• Multiply 4 by 5, resulting in 20.

For the term \(-3(x^2 - 2x + 1)\), we multiply each term by -3:

• Multiply -3 by \( x^2 \), resulting in \( -3x^2 \).
• Multiply -3 by \( -2x \), resulting in \( 6x \).
• Multiply -3 by 1, resulting in \( -3 \).

By applying the Distributive Property correctly, you make it easier to proceed with further simplifications.

Combining Like Terms

Combining like terms is essential in algebra as it helps to simplify expressions by merging terms that have the same variable part. Like terms are terms that involve the same variable raised to the same power. No matter what the coefficients are, if the variable part is identical, the terms can be combined into a single term. To combine like terms:

• Look for terms that match in variable parts and group them together.
• Add or subtract their coefficients.

Taking an example from the exercise:

• We first have \( 4x^2 - 3x^2 \): both these are \( x^2 \) terms. Combine them to get \( x^2 \).
• Next, \( -12x + 6x \) are both \( x \) terms. Combine them to result in \( -6x \).
• Finally, the constants 20 and -3 are paired up, resulting in the combined constant 17.

By combining like terms properly, you reduce the complexity of algebraic expressions, making them easier to handle.

Simplification of Expressions

Simplification is about transforming expressions into a simpler or more standardized form, which makes them easier to interpret and use for further calculations. In algebra, simplification typically involves two main steps: distributing and combining like terms. Here, you aim to create the simplest form of an expression by executing these steps accurately.Upon finishing the distribution and combining like terms from our example:

• You obtain the expression \( x^2 - 6x + 17 \).
• This is more concise and easier to manage since all operations are effectively condensed.
• Structure the expression neatly by listing the polynomial terms in descending order of power (i.e., start with \( x^2 \), followed by \( x \), and then any constant).

Proper simplification leads to a clean and clear representation of the problem, which not only aids in understanding and solving further mathematical procedures but also ensures your communication, especially in working out problems, remains precise and understandable.