Question

#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

Answer 1

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.