Problem 26 - Algebra and Trigonometry

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 constants

We begin by distributing the constants in front of the parentheses to each term inside the parentheses. This means we will multiply each term inside the parentheses by the constants in front of them. The expression is:\[4(x^2 - 3x + 5) - 3(x^2 - 2x + 1)\]Distributing the 4 gives us:\[4 \cdot x^2 - 4 \cdot 3x + 4 \cdot 5 = 4x^2 - 12x + 20\]Distributing the -3 gives us:\[-3 \cdot x^2 + -3 \cdot (-2x) + -3 \cdot 1 = -3x^2 + 6x - 3\]

Step 2: Combine like terms

Now that we have distributed the constants, we can combine the like terms.The expression becomes:\[4x^2 - 12x + 20 - 3x^2 + 6x - 3\]Combine the coefficients of like terms:- For the \(x^2\) terms: \(4x^2 - 3x^2 = x^2\)- For the \(x\) terms: \(-12x + 6x = -6x\)- For the constant terms: \(20 - 3 = 17\)

Step 3: Write the simplified expression

Combine the results from the like terms to write the simplified expression.The resulting expression is:\[x^2 - 6x + 17\]

Key Concepts

Polynomial Simplification

Polynomial simplification involves reducing a polynomial expression to its simplest form. This means getting rid of parentheses, combining like terms, and making the polynomial easier to understand and work with. In any given polynomial, you might see terms with variables and numbers combined through various mathematical operations.
For example: The expression \(4(x^2 - 3x + 5) - 3(x^2 - 2x + 1)\) can look complicated at first. By using simplification techniques, we can break it down into more manageable parts.
Here are a few key steps in polynomial simplification:

• Remove parentheses: This often involves the distributive property, which helps to eliminate the parentheses by multiplying each term within.
• Combine like terms: This involves adding or subtracting coefficients (numbers in front of variables) of the same degree.
• Finalize simplification: The last step is to collect all your like terms and write them one after another, usually ordered by decreasing powers of the variable.

Distributive Property

The distributive property is a fundamental algebraic principle used when simplifying expressions. It states that multiplying a single term by a group of terms inside a parenthesis is equivalent to multiplying that single term by each term separately and then adding the products.
For example, in the expression \(4(x^2 - 3x + 5)\), the 4 is distributed through each term in the parenthesis:

• 4 is multiplied by \(x^2\) giving \(4x^2\)
• 4 is multiplied by \(-3x\) giving \(-12x\)
• 4 is multiplied by 5 giving 20

This separate multiplication is helpful because it removes the parentheses and breaks down the expression into simpler parts.
It's important to note that if there is a subtraction sign in front, like \(-3(x^2 - 2x + 1)\), you need to multiply each term by \(-3\), not just 3. Thus, \(-3\) is distributed as follows:

• \(-3 \cdot x^2 = -3x^2\)
• \(-3 \cdot -2x = 6x\)
• \(-3 \cdot 1 = -3\)

Using the distributive property is a powerful tool in breaking down and rearranging expressions for simplification.

Combining Like Terms

Combining like terms is an essential step in the simplification process, and it helps in organizing and compressing an expression into a form that's easier to understand and solve. 'Like terms' are terms that have exactly the same variable parts. You can only combine terms that meet this criterion.
For example:

• Terms like \(x^2\), \(-3x^2\), and \(4x^2\) can be combined because they all have the variable \(x\) raised to the power of 2.
• Similarly, \(-12x\) and \(6x\) are like terms because they both have the variable \(x\) raised to the power of 1.

After you have applied the distributive property and identified all the like terms, you simply add or subtract their coefficients.

• With \(4x^2 - 3x^2\), you perform \(4 - 3\) to get \(x^2\).
• With \(-12x + 6x\), you perform \(-12 + 6\) to get \(-6x\).
• Finally, you handle numbers in the same way, so \(20 - 3\) becomes 17.

After combining, you conclude with a much simpler polynomial: \(x^2 - 6x + 17\). This makes the expression more manageable and clears the path for further solving if needed.