Solved) - Worksheet - adding and subtracting. polynomials ... - Free Printable

Problem Description:

The image shows a worksheet titled "Adding and Subtracting Polynomials." The task involves performing polynomial addition and subtraction for the given pairs of polynomials. Below, I will solve one example from each section (addition and subtraction) to demonstrate the process.

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Section 1: Adding Polynomials

#### Example Problem:
Problem 1: Add the polynomials \((2x^2 - 4x + 1)\) and \((x^2 - 9x - 5)\).

#### Solution:
1. Write down the polynomials to be added:
\[
(2x^2 - 4x + 1) + (x^2 - 9x - 5)
\]

2. Group like terms together:
- Terms with \(x^2\): \(2x^2\) and \(x^2\)
- Terms with \(x\): \(-4x\) and \(-9x\)
- Constant terms: \(1\) and \(-5\)

So, rewrite the expression as:
\[
(2x^2 + x^2) + (-4x - 9x) + (1 - 5)
\]

3. Combine like terms:
- For \(x^2\)-terms: \(2x^2 + x^2 = 3x^2\)
- For \(x\)-terms: \(-4x - 9x = -13x\)
- For constant terms: \(1 - 5 = -4\)

Therefore, the sum is:
\[
3x^2 - 13x - 4
\]

4. Final Answer:
\[
\boxed{3x^2 - 13x - 4}
\]

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Section 2: Subtracting Polynomials

#### Example Problem:
Problem 6: Subtract the polynomial \((8x^2 - x - 6)\) from \((2x^2 + 6x - 1)\).

#### Solution:
1. Write down the polynomials for subtraction:
\[
(2x^2 + 6x - 1) - (8x^2 - x - 6)
\]

2. Distribute the negative sign to the second polynomial:
\[
2x^2 + 6x - 1 - 8x^2 + x + 6
\]

3. Group like terms together:
- Terms with \(x^2\): \(2x^2\) and \(-8x^2\)
- Terms with \(x\): \(6x\) and \(x\)
- Constant terms: \(-1\) and \(6\)

So, rewrite the expression as:
\[
(2x^2 - 8x^2) + (6x + x) + (-1 + 6)
\]

4. Combine like terms:
- For \(x^2\)-terms: \(2x^2 - 8x^2 = -6x^2\)
- For \(x\)-terms: \(6x + x = 7x\)
- For constant terms: \(-1 + 6 = 5\)

Therefore, the result of the subtraction is:
\[
-6x^2 + 7x + 5
\]

5. Final Answer:
\[
\boxed{-6x^2 + 7x + 5}
\]

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General Explanation:

- Addition of Polynomials: Combine like terms by adding their coefficients.
- Subtraction of Polynomials: Distribute the negative sign to the polynomial being subtracted, then combine like terms.

If you need solutions for other specific problems from the worksheet, let me know!