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#1. Factor the given polynomials using difference of squares.

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Binomial:

4x2−36

Binomial:

9x2−25

Binomial:

9−25x2

Binomial:

81x2−49

Binomial:

4x2+36

1 answer

To factor the given binomials using the difference of squares, we can use the formula:

\[ a^2 - b^2 = (a - b)(a + b) \]

We will apply this to each binomial that represents a difference of squares.

  1. For \( 4x^2 - 36 \):

    Here, \( a^2 = 4x^2 \) and \( b^2 = 36 \).

    • \( a = 2x \)
    • \( b = 6 \)

    Thus, it factors as: \[ 4x^2 - 36 = (2x - 6)(2x + 6) \]

  2. For \( 9x^2 - 25 \):

    Here, \( a^2 = 9x^2 \) and \( b^2 = 25 \).

    • \( a = 3x \)
    • \( b = 5 \)

    Thus, it factors as: \[ 9x^2 - 25 = (3x - 5)(3x + 5) \]

  3. For \( 9 - 25x^2 \):

    We can rearrange it to \( - (25x^2 - 9) \), which gives us a difference of squares. Here, \( a^2 = 9 \) and \( b^2 = 25x^2 \).

    • \( a = 3 \)
    • \( b = 5x \)

    Thus, it factors as: \[ 9 - 25x^2 = -(25x^2 - 9) = (3 - 5x)(3 + 5x) \]

  4. For \( 81x^2 - 49 \):

    Here, \( a^2 = 81x^2 \) and \( b^2 = 49 \).

    • \( a = 9x \)
    • \( b = 7 \)

    Thus, it factors as: \[ 81x^2 - 49 = (9x - 7)(9x + 7) \]

  5. For \( 4x^2 + 36 \):

    This expression is not a difference of squares; it represents a sum of squares and cannot be factored using the difference of squares method.

So, the factored forms of the polynomials using the difference of squares are:

  • \( 4x^2 - 36 = (2x - 6)(2x + 6) \)
  • \( 9x^2 - 25 = (3x - 5)(3x + 5) \)
  • \( 9 - 25x^2 = (3 - 5x)(3 + 5x) \)
  • \( 81x^2 - 49 = (9x - 7)(9x + 7) \)
  • \( 4x^2 + 36 \) does not factor with this method.