Which pattern can be used to factor the high-degree polynomial 16x8−49x2?(1 point)

Question

Responses

(a+b)2=a2+2ab+b2
left parenthesis a plus b right parenthesis squared equals a squared plus 2 a b plus b squared

Neither the difference of squares nor the perfect square trinomial pattern can be used to factor 16x8−49x2.
Neither the difference of squares nor the perfect square trinomial pattern can be used to factor 16 x superscript 8 baseline minus 49 x squared .

(a−b)2=a2−2ab+b2
left parenthesis a minus b right parenthesis squared equals a squared minus 2 a b plus b squared

(a−b)(a+b)=a2−b2

Answer 1

To factor the polynomial \( 16x^8 - 49x^2 \), you can use the difference of squares pattern. The polynomial can be rewritten in a form that reveals this pattern.

First, notice that both terms, \( 16x^8 \) and \( 49x^2 \), are perfect squares:

• \( 16x^8 = (4x^4)^2 \)
• \( 49x^2 = (7)^2 \)

Thus, we can express the polynomial as: \[ 16x^8 - 49x^2 = (4x^4)^2 - (7)^2 \]

Now, we can apply the difference of squares formula, which states that: \[ a^2 - b^2 = (a - b)(a + b) \] In this case, let \( a = 4x^4 \) and \( b = 7 \). Using the difference of squares formula, we get: \[ 16x^8 - 49x^2 = (4x^4 - 7)(4x^4 + 7) \]

So, the correct response would be:

(a - b)(a + b) = a² - b²