Graphic Organizer - Adding Subtracting and Multiplying Poly | PDF - Free Printable

Problem Overview:

The image provides guidelines for adding, subtracting, and multiplying polynomials, along with examples for each operation. The task is to solve the given examples and explain the process step by step.

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1. Adding Polynomials

#### Guidelines:
- Combine like terms (terms with the same variable raised to the same power).
- Add coefficients of like terms.
- Exponents do not change.

#### Examples:

##### Example 1:
\[
(2x^2 - 4x + 4) + (-2x^2 - 5x + 4)
\]
- Combine like terms:
- \(2x^2 + (-2x^2) = 0x^2 = 0\)
- \(-4x + (-5x) = -9x\)
- \(4 + 4 = 8\)
- Result:
\[
0x^2 - 9x + 8 = -9x + 8
\]

##### Example 2:
\[
(7x^3 + 6x^2 - 2x) + (9x^2 - 4x + 3)
\]
- Combine like terms:
- \(7x^3\) (no other \(x^3\) term, so it remains as is)
- \(6x^2 + 9x^2 = 15x^2\)
- \(-2x + (-4x) = -6x\)
- \(3\) (no other constant term, so it remains as is)
- Result:
\[
7x^3 + 15x^2 - 6x + 3
\]

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2. Subtracting Polynomials

#### Guidelines:
- Switch the signs of all terms in the polynomial being subtracted.
- Then follow the rules for adding polynomials.

#### Examples:

##### Example 1:
\[
(2x^2 - 4x + 4) - (-2x^2 - 5x + 4)
\]
- Switch the signs of the second polynomial:
\[
-(-2x^2 - 5x + 4) = 2x^2 + 5x - 4
\]
- Now add the polynomials:
\[
(2x^2 - 4x + 4) + (2x^2 + 5x - 4)
\]
- Combine like terms:
- \(2x^2 + 2x^2 = 4x^2\)
- \(-4x + 5x = x\)
- \(4 + (-4) = 0\)
- Result:
\[
4x^2 + x
\]

##### Example 2:
\[
(7x^3 + 6x^2 - 2x) - (9x^2 - 4x + 3)
\]
- Switch the signs of the second polynomial:
\[
-(9x^2 - 4x + 3) = -9x^2 + 4x - 3
\]
- Now add the polynomials:
\[
(7x^3 + 6x^2 - 2x) + (-9x^2 + 4x - 3)
\]
- Combine like terms:
- \(7x^3\) (no other \(x^3\) term, so it remains as is)
- \(6x^2 + (-9x^2) = -3x^2\)
- \(-2x + 4x = 2x\)
- \(-3\) (no other constant term, so it remains as is)
- Result:
\[
7x^3 - 3x^2 + 2x - 3
\]

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3. Multiplying Polynomials

#### Guidelines:
- Use the distributive property to multiply each term of one polynomial by each term of the other polynomial.
- For binomials, the FOIL method can be used (First, Outer, Inner, Last).
- Combine like terms in the final result.

#### Examples:

##### Example 1:
\[
(2x^2 - 4x + 4)(-2x^2 - 5x + 4)
\]
- Distribute each term of the first polynomial to each term of the second polynomial:
\[
(2x^2)(-2x^2) + (2x^2)(-5x) + (2x^2)(4) + (-4x)(-2x^2) + (-4x)(-5x) + (-4x)(4) + (4)(-2x^2) + (4)(-5x) + (4)(4)
\]
- Calculate each product:
- \(2x^2 \cdot -2x^2 = -4x^4\)
- \(2x^2 \cdot -5x = -10x^3\)
- \(2x^2 \cdot 4 = 8x^2\)
- \(-4x \cdot -2x^2 = 8x^3\)
- \(-4x \cdot -5x = 20x^2\)
- \(-4x \cdot 4 = -16x\)
- \(4 \cdot -2x^2 = -8x^2\)
- \(4 \cdot -5x = -20x\)
- \(4 \cdot 4 = 16\)
- Combine like terms:
- \(x^4\) terms: \(-4x^4\)
- \(x^3\) terms: \(-10x^3 + 8x^3 = -2x^3\)
- \(x^2\) terms: \(8x^2 + 20x^2 - 8x^2 = 20x^2\)
- \(x\) terms: \(-16x - 20x = -36x\)
- Constant terms: \(16\)
- Result:
\[
-4x^4 - 2x^3 + 20x^2 - 36x + 16
\]

##### Example 2:
\[
(4x - 5)(3x + 7)
\]
- Use the FOIL method:
- First: \(4x \cdot 3x = 12x^2\)
- Outer: \(4x \cdot 7 = 28x\)
- Inner: \(-5 \cdot 3x = -15x\)
- Last: \(-5 \cdot 7 = -35\)
- Combine like terms:
- \(12x^2 + 28x - 15x - 35 = 12x^2 + 13x - 35\)
- Result:
\[
12x^2 + 13x - 35
\]

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Final Answers:

1. Adding Polynomials:
- Example 1: \(\boxed{-9x + 8}\)
- Example 2: \(\boxed{7x^3 + 15x^2 - 6x + 3}\)

2. Subtracting Polynomials:
- Example 1: \(\boxed{4x^2 + x}\)
- Example 2: \(\boxed{7x^3 - 3x^2 + 2x - 3}\)

3. Multiplying Polynomials:
- Example 1: \(\boxed{-4x^4 - 2x^3 + 20x^2 - 36x + 16}\)
- Example 2: \(\boxed{12x^2 + 13x - 35}\)

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These solutions follow the guidelines provided in the image and ensure clarity in each step.