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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) \( (\sqrt{6}-\sqrt{2}) \)

Ask by Parry Schofield. in Colombia
Mar 15,2025

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Tutor-Verified Answer

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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Extra Insights

Did you know that polynomials have been studied for centuries, with roots tracing back to ancient civilizations? The Babylonians were among the first to solve polynomial equations, laying the groundwork for algebra. Fast forward to today, where polynomial functions play crucial roles in fields like physics, economics, and engineering, helping to model everything from projectile motion to supply curves in economics. They’re the superheroes of mathematical modeling! If you’re working on factoring the polynomials listed, be on the lookout for common mistakes! One common error is not checking for a greatest common factor first before diving into more complex methods like synthetic division or the quadratic formula. Always double-check your work, especially signs when dealing with subtraction or negatives—one wrong sign can lead to a completely different polynomial!

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