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:
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Binomial: \(81 - 49x^2\)
- This can be factored as \( (9 - 7x)(9 + 7x) \) since \(81 = 9^2\) and \(49x^2 = (7x)^2\).
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Binomial: \(9x^2 - 25\)
- This can be factored as \( (3x - 5)(3x + 5) \) since \(9x^2 = (3x)^2\) and \(25 = 5^2\).
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Binomial: \(4x^2 + 36\)
- This cannot be factored using the difference of squares, as it is a sum of squares.
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Binomial: \(4x^2 - 36\)
- This can be factored as \( (2x - 6)(2x + 6) \) since \(4x^2 = (2x)^2\) and \(36 = 6^2\).
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Binomial: \(9 - 25x^2\)
- This can be factored as \( (3 - 5x)(3 + 5x) \) since \(9 = 3^2\) and \(25x^2 = (5x)^2\).
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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):
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Polynomial: \(3 + 5x^3 - 5x\)
- This polynomial cannot be factored using difference of squares.
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Polynomial: \(9x - 79x + 7\)
- Simplifying gives \( -70x + 7\), which cannot be factored using difference of squares.
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Polynomial: \(2x + 62x - 6\)
- Simplifying gives \(64x - 6\), which cannot be factored using difference of squares.
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Polynomial: \(3x + 53x - 5\)
- This simplifies to \(56x - 5\), which cannot be factored using difference of squares.
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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.