Answer
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 \( (2a^{n}+b)(2a^{n}-b) \)
n) \( x^{8}-y^{8} \) se factoriza como \( (x-y)(x+y)(x^{2}+y^{2})(x^{4}+y^{4}) \)
o) \( (\sqrt{6}-\sqrt{2}) \) no se puede factorizar.
Si necesitas más ayuda o aclaraciones sobre alguna de las factorizaciones, no dudes en preguntar.
Solution
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 \( x^{6}+x^{3}-2 \).
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 \( a^{2}-b^{3}+2 b^{3} x^{2}-2 a^{2} x^{2} \).
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 \( x^{4}+x-x^{3} y-y \).
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 \( x^{3}-x-x^{2} y+y \).
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 \( 6 x^{2}+x y-y^{2 \).
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 \( 4 a^{2 n}-b^{2 \).
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 \( m^{4}+m^{2} n^{2}+n^{4 \).
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 \( 2 \sqrt[3]{x^{2}}+5 \sqrt[3]{x}+2 \).
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:
\(\left(\sqrt[3]{x}+2\right)\left(2\sqrt[3]{x}+1\right)\)
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 \( 6 x^{2}+7 x y-3 y^{2 \).
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 \( a^{2}-7 a+12 \).
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 \( a^{2}+9 a+20 \).
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 \( a^{2}-6 a+9 \).
Factor the expression by following steps:
- step0: Factor:
\(a^{2}-6a+9\)
- step1: Factor the expression:
\(\left(a-3\right)^{2}\)
Factor the expression \( 15+14 x-8 x^{2 \).
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:
\(\left(5-2x\right)\left(3+4x\right)\)
Aquí están las factorizaciones de las expresiones proporcionadas:
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:
\[
(2a^{n}+b)(2a^{n}-b)
\]
n) \( x^{8}-y^{8} \) se factoriza como:
\[
(x-y)(x+y)(x^{2}+y^{2})(x^{4}+y^{4})
\]
o) \( (\sqrt{6}-\sqrt{2}) \) no se pudo factorizar.
Si necesitas más ayuda o aclaraciones sobre alguna de las factorizaciones, no dudes en preguntar.
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