Question

fatore a soma de quadrados:

a) a (elevado a 4) + 4
b) a (elevado a 4) + 1

Answer 1

Olá Rebeca, boa noite!

a)

$\\ \displaystyle \mathsf{a^4 + 4 =} \\\\ \mathsf{(a^2)^2 + (2^1)^2 =} \\\\ \mathsf{(a^2)^2 + (2)^2 =} \\\\ \mathsf{(a^2)^2 + (2)^2 + 2 \cdot (a^2) \cdot (2) - 2 \cdot (a^2) \cdot 2 =} \\\\ \mathsf{\left [(a^2)^2 + 2 \cdot (2^2) \cdot (2) + (2)^2 \right ] - 2 \cdot (a^2) \cdot 2 =} \\\\ \mathsf{\left[(a^2)+(2)\right]^2-4\cdot(a^2)=} \\\\ \mathsf{\left [ (a^2) + (2)\right ]^2 - 4a^2 =} \\\\ \mathsf{\left [ (a^2 + 2) + 2a \right ] \cdot \left [ (a^2 + 2) - 2a \right ] =} \\\\ \boxed{\mathsf{(a^2 + 2a + 2)(a^2 - 2a + 2)}}$

b)

$\\ \displaystyle \mathsf{a^4 + 1 =} \\\\ \mathsf{(a^2)^2 + (1)^2 =} \\\\ \mathsf{(a^2)^2 + (1)^2 + 2 \cdot (a^2) \cdot (1) - 2 \cdot (a^2) \cdot 1 =} \\\\ \mathsf{\left [(a^2)^2 + 2 \cdot (a^2) \cdot (1) + (1)^2 \right ] - 2 \cdot (a^2) \cdot 1 =} \\\\ \mathsf{\left [ (a^2) + (1)\right ]^2 - 2a^2 =} \\\\ \mathsf{\left [ (a^2 + 1) + \sqrt{2}a \right ] \cdot \left [ (a^2 + 1) - \sqrt{2}a \right ] =} \\\\ \boxed{\mathsf{(a^2 + a\sqrt{2} + 1)(a^2 - a\sqrt{2} + 1)}}$

 Obs.:

$\\ \mathsf{\bullet \qquad (a + b)^2 = (a)^2 + 2 \cdot (a) \cdot (b) + (b)^2} \\\\ \mathsf{\bullet \qquad a^2 - b^2 = (a + b) \cdot (a - b)}$