Problem 43 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(3 x^{2}+5 x-8\right)+\left(5 x^{2}+9 x+12\right)-\left(8 x^{2}-14\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(14x + 18\).

Step 1: Distribute the Minus Sign

First, recognize that subtraction of polynomials involves distributing the negative sign across the terms of the polynomial being subtracted. This means we need to change the signs of the terms inside the last set of parentheses. Thus, \[-(8x^{2} - 14) = -8x^{2} + 14.\]

Step 2: Simplify the Expression

Now rewrite the original expression, applying the result of the distribution step: \[(3x^{2} + 5x - 8) + (5x^{2} + 9x + 12) + (-8x^{2} + 14).\]Replace the subtraction with additional terms from Step 1.

Step 3: Combine Like Terms

Group the terms with the same degree: - Combine \(x^{2}\) terms: \[3x^{2} + 5x^{2} - 8x^{2} = 0x^{2}.\]- Combine \(x\) terms: \[5x + 9x = 14x.\]- Combine constant terms: \[-8 + 12 + 14 = 18.\]

Step 4: Write Final Simplified Expression

Gather the results of the like terms: The polynomial simplifies to \[14x + 18.\]

Key Concepts

Adding Polynomials

When we talk about adding polynomials, we're simply referring to the process of combining two or more polynomial expressions. Each polynomial is typically a sum of terms, where each term is a coefficient (a number) multiplied by a variable raised to an exponent. To add polynomials, we place them side by side and look for terms that can be combined. The easiest way to approach this is:

• Align each polynomial according to the power of its variables.
• Add coefficients in front of like terms. This means terms with the same variable raised to the same exponent.
• Make sure to keep variable terms with their appropriate exponents, as only like terms can be combined.

For example, if you have \[ (3x^{2} + 4x + 5) + (5x^{2} + 7x + 2) \]then you should:

• Add the \(x^2\) terms: \[ 3x^{2} + 5x^{2} = 8x^{2} \]
• Add the \(x\) terms:\[ 4x + 7x = 11x \]
• Add the constant terms:\[ 5 + 2 = 7 \]

Thus, the final polynomial after addition will be:\[ 8x^{2} + 11x + 7. \] This process ensures the polynomials are neatly combined into a single expression.

Subtracting Polynomials

Subtracting polynomials involves a bit more than just subtraction. It requires distributing the negative sign across the entire polynomial being subtracted to ensure that every term changes its sign. This distribution is crucial because neglecting to do so might lead to incorrect results.Here's how it works:

• Begin by changing the signs of the terms in the polynomial that is being subtracted. Convert each "positive" term to its "negative" equivalent and each "negative" term to its "positive" equivalent.
• Once all signs are flipped, align the expression with the other polynomials, similar to the addition process.
• Now, perform addition, combining like terms as though you were adding polynomials.

Suppose we have\[ (3x^2 + 5x - 8) - (2x^2 + 7) \].Perform the following:

• Distribute the negative sign:\[ - (2x^2 + 7) = -2x^2 - 7 \].
• Rewrite the expression as:\[ 3x^2 + 5x - 8 + (-2x^2 - 7) \].
• Combine like terms for a final expression:\[ (3x^2 + (-2x^2)) + 5x + (-8 - 7) = x^2 + 5x - 15 \].

By distributing the negative sign, this ensures each term from the second polynomial affects the expression correctly.

Combining Like Terms

When you're working with polynomials, it's essential to recognize and combine like terms. Like terms are terms that have the same variable raised to the same power. This is important because only like terms can be logically added or subtracted without changing the integrity of the expression.To combine like terms:

• Identify terms with the same variable and exponent.
• Add or subtract the coefficients of these terms.

For instance, if you are given:\[ 7x^2 + 4x - 3x^2 + 5 - 6 \].You should:

• Combine \(x^2\) terms:\[ 7x^2 - 3x^2 = 4x^2 \].
• Keep the \(x\) term alone since there's only one:\[ 4x \].
• Combine the constant terms:\[ 5 - 6 = -1 \].

By simplifying in this way, your final polynomial becomes:\[ 4x^2 + 4x - 1 \].This simplification step is crucial to make expressions manageable and easier to understand or utilize in further calculations.