Problem 22 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result after simplification is \(x^2 + 4x + 6\).

Step 1: Distribute the Negative Sign

To simplify the expression, distribute the negative sign across the second polynomial inside the parentheses. This changes the signs of each term in the second polynomial: \[ -(2x^2 - 3x - 5) = -2x^2 + 3x + 5 \]

Step 2: Rewrite the Expression

Rewrite the original expression by combining terms from each polynomial, including the effects of distributing the negative sign:\[ (3x^2 + x + 1) + (-2x^2 + 3x + 5) \]

Step 3: Combine Like Terms

Combine the like terms from the two polynomials to simplify. Add the coefficients for the \(x^2\) terms, \(x\) terms, and constant terms separately:- \(x^2\) terms: \((3x^2) + (-2x^2) = x^2\)- \(x\) terms: \((x) + (3x) = 4x\)- Constant terms: \((1) + (5) = 6\)This results in the expression:\[ x^2 + 4x + 6 \]

Key Concepts

Combining Like Terms

When working with polynomials, combining like terms is an essential skill. Like terms are terms that include the same variable raised to the same power. For instance, consider terms like \(3x^2\) and \(-2x^2\), both of which have the variable \(x\) raised to the power of 2. To combine terms like these, you need to add or subtract their coefficients.

• The process simplifies the expression by merging similar items into a single term.
• It often involves grouping and reorganizing terms to make addition or subtraction more straightforward.
• Always be mindful of the sign in front of each term, as this indicates whether you should add or subtract.

For example, if you have terms \(5x\) and \(2x\), you simply add the coefficients (5 and 2) to get \(7x\). When completed, a polynomial is in a cleaner, more consolidated form, making further operations easier to execute.

Polynomial Subtraction

Polynomial subtraction can be a bit tricky, particularly due to the involvement of negative signs. The main step is distributing the negative sign over the entire polynomial being subtracted. This effectively changes addition into subtraction and must be done carefully.

• Consider the expression \((3x^2 + x + 1) - (2x^2 - 3x - 5)\).
• Distribute the negative sign to each term: change \(2x^2\) to \(-2x^2\), \(-3x\) to \(+3x\), and \(-5\) to \(+5\).
• Now, rewrite total expression as \((3x^2 + x + 1) + (-2x^2 + 3x + 5)\).

This step fundamentally converts subtraction into adding a negative, which then allows you to naturally proceed to add like terms. The key is handling negative signs accurately to ensure each term is correctly represented.

Distributive Property

The distributive property is crucial when working with polynomial expressions, especially during subtraction. It allows you to multiply a single term across terms inside a bracket. This is useful for both distributing constants and dealing with subtraction.

• Think of the distributive property as a way to "distribute" a number or a sign (such as a negative) to all elements inside a parenthesis.
• In subtraction, the distributive property aids in altering the signs of a polynomial being subtracted.
• For example, applying \(-(2x^2 - 3x - 5)\) gives \(-2x^2 + 3x + 5\).

By transforming subtraction into addition of negatives, the distributive property ensures smooth and error-free polynomial manipulation. Understanding how and when to apply this property simplifies subtraction and enhances algebraic efficiency.