5 Ejercicios resueltos: Factorización por agrupación – Nivel difícil

Desarrollo:

$$ 2ax + 2bx – ay – by – az – bz $$

$$ a \left(2x – y – z \right) + b \left(2x – y – z \right) $$

$$ \left(a + b \right) \left(2x – y – z \right) $$

Respuesta:

$$ \boxed{\left(a + b \right) \left(2x – y – z \right)} $$

Desarrollo:

$$ p^{4} + p^{2}q^{2} + p^{2}r^{2} + 2p^{2}q + 2q^{3} + 2qr^{2} + p^{2}r + q^{2}r + r^{3} $$

$$ p^{2} \left(p^{2} + q^{2} + r^{2} \right) + 2q \left(p^{2} + q^{2} + r^{2} \right) + r \left(p^{2} + q^{2} + r^{2} \right) $$

$$ \left(p^{2} + r + 2q \right) \left(p^{2} + q^{2} + r^{2} \right) $$

Respuesta:

$$ \boxed{\left(p^{2} + r + 2q \right) \left(p^{2} + q^{2} + r^{2} \right)} $$

Desarrollo:

$$ x^{n+3} + x^{4} + x^{n} + x $$

$$ x^{n} x^{3} + x^{4} + x^{n} + x $$

Agrupamos

$$ x^{n} x^{3} + x^{n} + x^{4} + x $$

$$ x^{n} \left(x^{3} + 1 \right) + x \left(x^{3} + 1 \right) $$

$$ \left(x^{n} + x \right) \left(x^{3} + 1 \right) $$

Respuesta:

$$ \boxed{\left(x^{n} + x \right) \left(x^{3} + 1 \right)} $$

Desarrollo:

$$ x^{5} + 2x^{4} – x^{3} – 2^{2} $$

$$ x^{2} \left(x^{3} + 2x^{2} – x – 2 \right) $$

$$ x^{2} \left[x^{3} + 2x^{2} – x – 2 \right] $$

$$ x^{2} \left[x^{2} \left(x + 2 \right) – x – 2 \right] $$

$$ x^{2} \left[x^{2} \left(x + 2 \right) – \left(x + 2 \right) \right] $$

$$ x^{2} \left[ \left(x^{2} – 1 \right) \left(x + 2 \right) \right] $$

Respuesta:

$$ \boxed{x^{2} \left(x^{2} – 1 \right) \left(x + 2 \right)} $$

Desarrollo:

$$ xy^{3} + xyz^{3} + y^{2}z + z^{4} $$

$$ xy \left(y^{2} + z^{3} \right) + z \left(y^{2} + z^{3} \right) $$

$$ \left(xy + z \right) \left(y^{2} + z^{3} \right) $$

Respuesta:

$$ \boxed{\left(xy + z \right) \left(y^{2} + z^{3} \right)} $$