Question

#1. Factor the given polynomials using difference of squares.
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cannot be factored
Binomial:



81−49x2

Binomial:



9x2−25

Binomial:



4x2+36

Binomial:



4x2−36

Binomial:



9−25x2

Binomial:



81x2−49

3+5x3−5x
9x−79x+7
2x+62x−6
3x+53x−5
9+7x9−7x

Answer 1

To factor the given polynomials using the difference of squares, we will identify binomials of the form \(a^2 - b^2\), which can be factored as \((a - b)(a + b)\).

Let's go through the provided binomials:

1. **Binomial: \(81 - 49x^2\)**
- This can be factored as \( (9 - 7x)(9 + 7x) \) since \(81 = 9^2\) and \(49x^2 = (7x)^2\).

2. **Binomial: \(9x^2 - 25\)**
- This can be factored as \( (3x - 5)(3x + 5) \) since \(9x^2 = (3x)^2\) and \(25 = 5^2\).

3. **Binomial: \(4x^2 + 36\)**
- This cannot be factored using the difference of squares, as it is a sum of squares.

4. **Binomial: \(4x^2 - 36\)**
- This can be factored as \( (2x - 6)(2x + 6) \) since \(4x^2 = (2x)^2\) and \(36 = 6^2\).

5. **Binomial: \(9 - 25x^2\)**
- This can be factored as \( (3 - 5x)(3 + 5x) \) since \(9 = 3^2\) and \(25x^2 = (5x)^2\).

6. **Binomial: \(81x^2 - 49\)**
- This can be factored as \( (9x - 7)(9x + 7) \) since \(81x^2 = (9x)^2\) and \(49 = 7^2\).

Now, for the polynomials (not binomials):

1. **Polynomial: \(3 + 5x^3 - 5x\)**
- This polynomial cannot be factored using difference of squares.

2. **Polynomial: \(9x - 79x + 7\)**
- Simplifying gives \( -70x + 7\), which cannot be factored using difference of squares.

3. **Polynomial: \(2x + 62x - 6\)**
- Simplifying gives \(64x - 6\), which cannot be factored using difference of squares.

4. **Polynomial: \(3x + 53x - 5\)**
- This simplifies to \(56x - 5\), which cannot be factored using difference of squares.

5. **Polynomial: \(9 + 7x^9 - 7x\)**
- Simplifying gives \(7x^9 - 7x + 9\), which cannot be factored using difference of squares.

### Summary of Factors

- \( 81 - 49x^2 = (9 - 7x)(9 + 7x) \)
- \( 9x^2 - 25 = (3x - 5)(3x + 5) \)
- \( 4x^2 + 36 \): cannot be factored
- \( 4x^2 - 36 = (2x - 6)(2x + 6) \)
- \( 9 - 25x^2 = (3 - 5x)(3 + 5x) \)
- \( 81x^2 - 49 = (9x - 7)(9x + 7) \)
- Other polynomials cannot be factored using difference of squares.