Question

Which pattern can be used to factor the high-degree polynomial  16x8−49x2 ?(1 point)Responses(a+b)2=a2+2ab+b2left parenthesis a plus b right parenthesis squared equals a squared plus 2 a b plus b squared(a−b)(a+b)=a2−b2left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis equals a squared minus b squared(a−b)2=a2−2ab+b2left parenthesis a minus b right parenthesis squared equals a squared minus 2 a b plus b squaredNeither 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 .

Answer 1

To factor the polynomial \( 16x^8 - 49x^2 \), we can recognize that it resembles the form of a difference of squares. The difference of squares pattern is expressed as:

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

In this case, we can rewrite \( 16x^8 - 49x^2 \) as follows:

1. Identify \( a \) and \( b \):

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

2. Now we apply the difference of squares formula:

\[ 16x^8 - 49x^2 = (4x^4)^2 - (7x)^2 = (4x^4 - 7x)(4x^4 + 7x) \]

Thus, the original expression can be factored into:

\[ (4x^4 - 7x)(4x^4 + 7x) \]

Therefore, the correct response indicating the pattern used to factor \( 16x^8 - 49x^2 \) is:

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

So, the correct answer would be:

\[ (a-b)(a+b)=a^2-b^2 \quad \text{(the difference of squares)} \]