5. Factorice completamente las siguientes expresiones. 1 a) $$ x^{4}+x-x^{3} y-y $$ b) $$ x^{3}-x-x^{2} y+y $$ c) $$ 6 x^{2}+x y-y^{2} $$ d) $$ a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} $$ e) $$ a^{2}+9 a+20 $$ f) $$ a^{2}-7 a+12 $$ g) $$ a^{2}-6 a+9 $$ h) $$ 6 x^{2}-x-2 $$ i) $$ 6 x^{2}+7 x y-3 y^{2} $$ j) $$ m^{4}+m^{2} n^{2}+n^{4} $$ k) $$ 15+14 x-8 x^{2} $$ l) $$ x^{6}+x^{3}-2 $$ m) $$ 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 $$ n) $$ 4 a^{2 n}-b^{2} $$ n) $$ x^{8}-y^{8} $$ o) $$ (\sqrt{6}-\sqrt{2}) $$

Question

UpStudy AI · Accepted Answer

Factor the expression by following steps:
- step0: Factor:
  $$a^{2}-b^{3}+2b^{3}x^{2}-2a^{2}x^{2}$$
- step1: Calculate:
  $$2b^{3}x^{2}-b^{3}-2a^{2}x^{2}+a^{2}$$
- step2: Rewrite the expression:
  $$b^{3}\times 2x^{2}-b^{3}-a^{2}\times 2x^{2}+a^{2}$$
- step3: Factor the expression:
  $$b^{3}\left(2x^{2}-1\right)-a^{2}\left(2x^{2}-1\right)$$
- step4: Factor the expression:
  $$\left(b^{3}-a^{2}\right)\left(2x^{2}-1\right)$$
Factor the expression $$ 6 x^{2}-x-2 $$.
Factor the expression by following steps:
- step0: Factor:
  $$6x^{2}-x-2$$
- step1: Rewrite the expression:
  $$6x^{2}+\left(-4+3\right)x-2$$
- step2: Calculate:
  $$6x^{2}-4x+3x-2$$
- step3: Rewrite the expression:
  $$2x\times 3x-2x\times 2+3x-2$$
- step4: Factor the expression:
  $$2x\left(3x-2\right)+3x-2$$
- step5: Factor the expression:
  $$\left(2x+1\right)\left(3x-2\right)$$
Factor the expression $$ x^{3}-x-x^{2} y+y $$.
Factor the expression by following steps:
- step0: Factor:
  $$x^{3}-x-x^{2}y+y$$
- step1: Calculate:
  $$x^{3}-x-yx^{2}+y$$
- step2: Rewrite the expression:
  $$x\times x^{2}-x-yx^{2}+y$$
- step3: Factor the expression:
  $$x\left(x^{2}-1\right)-y\left(x^{2}-1\right)$$
- step4: Factor the expression:
  $$\left(x-y\right)\left(x^{2}-1\right)$$
- step5: Factor the expression:
  $$\left(x-y\right)\left(x+1\right)\left(x-1\right)$$
Factor the expression $$ a^{2}-7 a+12 $$.
Factor the expression by following steps:
- step0: Factor:
  $$a^{2}-7a+12$$
- step1: Rewrite the expression:
  $$a^{2}+\left(-3-4\right)a+12$$
- step2: Calculate:
  $$a^{2}-3a-4a+12$$
- step3: Rewrite the expression:
  $$a\times a-a\times 3-4a+4\times 3$$
- step4: Factor the expression:
  $$a\left(a-3\right)-4\left(a-3\right)$$
- step5: Factor the expression:
  $$\left(a-4\right)\left(a-3\right)$$
Factor the expression $$ x^{4}+x-x^{3} y-y $$.
Factor the expression by following steps:
- step0: Factor:
  $$x^{4}+x-x^{3}y-y$$
- step1: Evaluate:
  $$x^{4}-x^{3}y+x-y$$
- step2: Calculate:
  $$x^{4}+x^{3}-x^{3}y-x^{2}y-x^{3}-x^{2}+x^{2}y+xy+x^{2}+x-yx-y$$
- step3: Rewrite the expression:
  $$x^{2}\times x^{2}+x^{2}\times x-x^{2}yx-x^{2}y-x\times x^{2}-x\times x+xyx+xy+x^{2}+x-yx-y$$
- step4: Factor the expression:
  $$x^{2}\left(x^{2}+x-yx-y\right)-x\left(x^{2}+x-yx-y\right)+x^{2}+x-yx-y$$
- step5: Factor the expression:
  $$\left(x^{2}-x+1\right)\left(x^{2}+x-yx-y\right)$$
- step6: Factor the expression:
  $$\left(x^{2}-x+1\right)\left(x-y\right)\left(x+1\right)$$
Factor the expression $$ a^{2}+9 a+20 $$.
Factor the expression by following steps:
- step0: Factor:
  $$a^{2}+9a+20$$
- step1: Rewrite the expression:
  $$a^{2}+\left(5+4\right)a+20$$
- step2: Calculate:
  $$a^{2}+5a+4a+20$$
- step3: Rewrite the expression:
  $$a\times a+a\times 5+4a+4\times 5$$
- step4: Factor the expression:
  $$a\left(a+5\right)+4\left(a+5\right)$$
- step5: Factor the expression:
  $$\left(a+4\right)\left(a+5\right)$$
Factor the expression $$ 6 x^{2}+x y-y^{2 $$.
Factor the expression by following steps:
- step0: Factor:
  $$6x^{2}+xy-y^{2}$$
- step1: Rewrite the expression:
  $$6x^{2}+\left(3-2\right)xy-y^{2}$$
- step2: Calculate:
  $$6x^{2}+3xy-2yx-y^{2}$$
- step3: Rewrite the expression:
  $$3x\times 2x+3xy-y\times 2x-y\times y$$
- step4: Factor the expression:
  $$3x\left(2x+y\right)-y\left(2x+y\right)$$
- step5: Factor the expression:
  $$\left(3x-y\right)\left(2x+y\right)$$
Factor the expression $$ x^{6}+x^{3}-2 $$.
Factor the expression by following steps:
- step0: Factor:
  $$x^{6}+x^{3}-2$$
- step1: Calculate:
  $$x^{6}+2x^{3}+x^{5}+2x^{2}+x^{4}+2x-x^{5}-2x^{2}-x^{4}-2x-x^{3}-2$$
- step2: Rewrite the expression:
  $$x\times x^{5}+x\times 2x^{2}+x\times x^{4}+x\times 2x+x\times x^{3}+x\times 2-x^{5}-2x^{2}-x^{4}-2x-x^{3}-2$$
- step3: Factor the expression:
  $$x\left(x^{5}+2x^{2}+x^{4}+2x+x^{3}+2\right)-\left(x^{5}+2x^{2}+x^{4}+2x+x^{3}+2\right)$$
- step4: Factor the expression:
  $$\left(x-1\right)\left(x^{5}+2x^{2}+x^{4}+2x+x^{3}+2\right)$$
- step5: Factor the expression:
  $$\left(x-1\right)\left(x^{2}+x+1\right)\left(x^{3}+2\right)$$
Factor the expression $$ 6 x^{2}+7 x y-3 y^{2 $$.
Factor the expression by following steps:
- step0: Factor:
  $$6x^{2}+7xy-3y^{2}$$
- step1: Rewrite the expression:
  $$6x^{2}+\left(9-2\right)xy-3y^{2}$$
- step2: Calculate:
  $$6x^{2}+9xy-2yx-3y^{2}$$
- step3: Rewrite the expression:
  $$3x\times 2x+3x\times 3y-y\times 2x-y\times 3y$$
- step4: Factor the expression:
  $$3x\left(2x+3y\right)-y\left(2x+3y\right)$$
- step5: Factor the expression:
  $$\left(3x-y\right)\left(2x+3y\right)$$
Factor the expression $$ m^{4}+m^{2} n^{2}+n^{4 $$.
Factor the expression by following steps:
- step0: Factor:
  $$m^{4}+m^{2}n^{2}+n^{4}$$
- step1: Calculate:
  $$m^{4}-m^{3}n+m^{2}n^{2}+m^{3}n-m^{2}n^{2}+mn^{3}+n^{2}m^{2}-n^{3}m+n^{4}$$
- step2: Rewrite the expression:
  $$m^{2}\times m^{2}-m^{2}\times mn+m^{2}n^{2}+mnm^{2}-mnmn+mn\times n^{2}+n^{2}m^{2}-n^{2}mn+n^{2}\times n^{2}$$
- step3: Factor the expression:
  $$m^{2}\left(m^{2}-mn+n^{2}\right)+mn\left(m^{2}-mn+n^{2}\right)+n^{2}\left(m^{2}-mn+n^{2}\right)$$
- step4: Factor the expression:
  $$\left(m^{2}+mn+n^{2}\right)\left(m^{2}-mn+n^{2}\right)$$
Factor the expression $$ 15+14 x-8 x^{2} $$.
Factor the expression by following steps:
- step0: Factor:
  $$15+14x-8x^{2}$$
- step1: Rewrite the expression:
  $$15+\left(20-6\right)x-8x^{2}$$
- step2: Calculate:
  $$15+20x-6x-8x^{2}$$
- step3: Rewrite the expression:
  $$5\times 3+5\times 4x-2x\times 3-2x\times 4x$$
- step4: Factor the expression:
  $$5\left(3+4x\right)-2x\left(3+4x\right)$$
- step5: Factor the expression:
  $$\left(5-2x\right)\left(3+4x\right)$$
Factor the expression $$ a^{2}-6 a+9 $$.
Factor the expression by following steps:
- step0: Factor:
  $$a^{2}-6a+9$$
- step1: Factor the expression:
  $$\left(a-3\right)^{2}$$
Factor the expression $$ 4 a^{2 n}-b^{2 $$.
Factor the expression by following steps:
- step0: Factor:
  $$4a^{2n}-b^{2}$$
- step1: Evaluate:
  $$4\left(a^{n}\right)^{2}-b^{2}$$
- step2: Factor the expression:
  $$\left(2a^{n}+b\right)\left(2a^{n}-b\right)$$
Factor the expression $$ x^{8}-y^{8 $$.
Factor the expression by following steps:
- step0: Factor:
  $$x^{8}-y^{8}$$
- step1: Rewrite the expression:
  $$\left(x^{4}\right)^{2}-\left(y^{4}\right)^{2}$$
- step2: Factor the expression:
  $$\left(x^{4}-y^{4}\right)\left(x^{4}+y^{4}\right)$$
- step3: Evaluate:
  $$\left(x-y\right)\left(x+y\right)\left(x^{2}+y^{2}\right)\left(x^{4}+y^{4}\right)$$
Factor the expression $$ 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 $$.
Factor the expression by following steps:
- step0: Factor:
  $$2\sqrt[3]{x^{2}}+5\sqrt[3]{x}+2$$
- step1: Evaluate:
  $$2x^{\frac{2}{3}}+5x^{\frac{1}{3}}+2$$
- step2: Rewrite the expression:
  $$2x^{\frac{2}{3}}+\left(1+4\right)x^{\frac{1}{3}}+2$$
- step3: Calculate:
  $$2x^{\frac{2}{3}}+x^{\frac{1}{3}}+4x^{\frac{1}{3}}+2$$
- step4: Rewrite the expression:
  $$x^{\frac{1}{3}}\times 2x^{\frac{1}{3}}+x^{\frac{1}{3}}+2\times 2x^{\frac{1}{3}}+2$$
- step5: Factor the expression:
  $$x^{\frac{1}{3}}\left(2x^{\frac{1}{3}}+1\right)+2\left(2x^{\frac{1}{3}}+1\right)$$
- step6: Factor the expression:
  $$\left(x^{\frac{1}{3}}+2\right)\left(2x^{\frac{1}{3}}+1\right)$$
- step7: Rewrite the expression:
  $$\left(\sqrt[3]{x}+2\right)\left(2\sqrt[3]{x}+1\right)$$
Aquí están las factorizaciones completas de las expresiones dadas:

a) $$ x^{4}+x-x^{3} y-y $$ se factoriza como:
$$
(x^{2}-x+1)(x-y)(x+1)
$$

b) $$ x^{3}-x-x^{2} y+y $$ se factoriza como:
$$
(x-y)(x+1)(x-1)
$$

c) $$ 6 x^{2}+x y-y^{2} $$ se factoriza como:
$$
(3x-y)(2x+y)
$$

d) $$ a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} $$ se factoriza como:
$$
(b^{3}-a^{2})(2x^{2}-1)
$$

e) $$ a^{2}+9 a+20 $$ se factoriza como:
$$
(a+4)(a+5)
$$

f) $$ a^{2}-7 a+12 $$ se factoriza como:
$$
(a-4)(a-3)
$$

g) $$ a^{2}-6 a+9 $$ se factoriza como:
$$
(a-3)^{2}
$$

h) $$ 6 x^{2}-x-2 $$ se factoriza como:
$$
(2x+1)(3x-2)
$$

i) $$ 6 x^{2}+7 x y-3 y^{2} $$ se factoriza como:
$$
(3x-y)(2x+3y)
$$

j) $$ m^{4}+m^{2} n^{2}+n^{4} $$ se factoriza como:
$$
(m^{2}+mn+n^{2})(m^{2}-mn+n^{2})
$$

k) $$ 15+14 x-8 x^{2} $$ se factoriza como:
$$
(5-2x)(3+4x)
$$

l) $$ x^{6}+x^{3}-2 $$ se factoriza como:
$$
(x-1)(x^{2}+x+1)(x^{3}+2)
$$

m) $$ 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 $$ se factoriza como:
$$
(\sqrt[3]{x}+2)(2\sqrt[3]{x}+1)
$$

n) $$ 4 a^{2 n}-b^{2} $$ se factoriza como:
$$
(2a^{n}+b)(2a^{n}-b)
$$

o) $$ x^{8}-y^{8} $$ se factoriza como:
$$
(x-y)(x+y)(x^{2}+y^{2})(x^{4}+y^{4})
$$

p) $$ (\sqrt{6}-\sqrt{2}) $$ no se puede factorizar más allá de su forma actual.

Si necesitas más ayuda o aclaraciones sobre alguna de las factorizaciones, no dudes en preguntar.
a) $$ x^{4}+x-x^{3} y-y = (x^{2}-x+1)(x-y)(x+1) $$ b) $$ x^{3}-x-x^{2} y+y = (x-y)(x+1)(x-1) $$ c) $$ 6 x^{2}+x y-y^{2} = (3x-y)(2x+y) $$ d) $$ a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} = (b^{3}-a^{2})(2x^{2}-1) $$ e) $$ a^{2}+9 a+20 = (a+4)(a+5) $$ f) $$ a^{2}-7 a+12 = (a-4)(a-3) $$ g) $$ a^{2}-6 a+9 = (a-3)^{2} $$ h) $$ 6 x^{2}-x-2 = (2x+1)(3x-2) $$ i) $$ 6 x^{2}+7 x y-3 y^{2} = (3x-y)(2x+3y) $$ j) $$ m^{4}+m^{2} n^{2}+n^{4} = (m^{2}+mn+n^{2})(m^{2}-mn+n^{2}) $$ k) $$ 15+14 x-8 x^{2} = (5-2x)(3+4x) $$ l) $$ x^{6}+x^{3}-2 = (x-1)(x^{2}+x+1)(x^{3}+2) $$ m) $$ 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 = (\sqrt[3]{x}+2)(2\sqrt[3]{x}+1) $$ n) $$ 4 a^{2 n}-b^{2} = (2a^{n}+b)(2a^{n}-b) $$ o) $$ x^{8}-y^{8} = (x-y)(x+y)(x^{2}+y^{2})(x^{4}+y^{4}) $$ p) $$ (\sqrt{6}-\sqrt{2}) $$ no se puede factorizar más.

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