Which pattern can be used to factor the high-degree polynomial 16x^8-49x^2?

Question

Which pattern can be used to factor the high-degree polynomial 16x^8-49x^2?
a. Neither the difference of squares nor the perfect square trinomial pattern can be used
b. (a-b)(a+b)=a^2-b^2
c. (a-b)^2=a^2-2ab+b^2
d. (a+b)^2=a^2+2ab+b^2

GPT-4o mini · Answer

To factor the given polynomial $ 16x^8 - 49x^2 $, we can first notice that it can be rewritten as: $$ 16x^8 - 49x^2 = (16x^8) - (49x^2) $$ This expression is indeed a difference of squares, because it can be expressed as $ a^2 - b^2 $ where: $$ a = (4x^4) \quad \text{and} \quad b = (7x) $$ Thus, we recognize that: $$ 16x^8 - 49x^2 = (4x^4)^2 - (7x)^2 $$ Using the difference of squares factoring pattern $ a^2 - b^2 = (a - b)(a + b) $, we can factor this polynomial as follows: $$ 16x^8 - 49x^2 = (4x^4 - 7x)(4x^4 + 7x) $$ Thus, the correct answer is option **b**: $(a-b)(a+b)=a^2-b^2$, as we used the difference of squares to factor the polynomial.