Question

factor the given polynomial.
\\16t^4 - 1\\

Explanation:

Identify the expression

The given polynomial is:

$$ 16t^4 - 1 $$

Apply difference of squares

Using the Polynomial Factoring knowledge point

$$ 16t^4 - 1 = (4t^2)^2 - 1^2 = (4t^2 - 1)(4t^2 + 1) $$

Factor completely

The factor \((4t^2 - 1)\) is itself a difference of squares:

$$ 4t^2 - 1 = (2t)^2 - 1^2 = (2t - 1)(2t + 1) $$

The factor \((4t^2 + 1)\) is a sum of squares and cannot be factored further over the real numbers.

Combine all factors

Substituting the factored form back into the expression:

$$ 16t^4 - 1 = (2t - 1)(2t + 1)(4t^2 + 1) $$

Answer:

Factor the given polynomial.
\(16t^4 - 1 =\) <blank>\((2t - 1)(2t + 1)(4t^2 + 1)\)</blank>