Toán Lớp 8: Phân tích đa thức thành nhân tử.
1. $x^{5}+x+1$
2. $x^{8}+x+1$
3. $x^{11}+x+1$
4. $x^{5}+x^{4}+1$
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Toán Lớp 8: Phân tích đa thức thành nhân tử.
1. $x^{5}+x+1$
2. $x^{8}+x+1$
3. $x^{11}+x+1$
4. $x^{5}+x^{4}+1$
Comments ( 1 )
$\text{ @HV }$
a) x^5 + x + 1
= x^5 + x^4 + x^3 – x^4 – x^3 – x^2 + x^2 + x + 1
= (x^5 + x^4 + x^3) – (x^4 + x^3 + x^2) + (x^2 + x + 1)
= x^3. (x^2 + x + 1) – x^2. (x^2 + x + 1) + 1. (x^2 + x + 1)
= (x^3 – x^2 + 1). (x^2 + x + 1)
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b) x^8 + x + 1
= x^8 + x^7 + x^6 – x^7 – x^6 – x^5 + x^5 + x^4 + x^3 – x^4 – x^3 – x^2 + x^2 + x + 1
= (x^8 + x^7 + x^6) – (x^7 + x^6 + x^5) + (x^5 + x^4 + x^3) – (x^4 + x^3 + x^2) + (x^2 + x + 1)
= x^6. (x^2 + x + 1) – x^5. (x^2 + x + 1) + x^3. (x^2 + x + 1) – x^2. (x^2 + x + 1) + 1. (x^2 + x + 1)
= (x^2 + x + 1). (x^6 – x^5 + x^3 – x^2 + 1)
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c) x^11 + x + 1
= x^11 + x^10 + x^9 – x^10 – x^9 – x^8 + x^8 + x^7 + x^6 – x^7 – x^6 – x^5 + x^5 + x^4 + x^3 – x^4 – x^3 – x^2 + x^2 + x + 1
= (x^11 + x^10 + x^9) – (x^10 + x^9 + x^8) + (x^8 + x^7 + x^6) – (x^7 + x^6 + x^5) + (x^5 + x^4 + x^3) – (x^4 + x^3 + x^2) + (x^2 + x + 1)
= x^9. (x^2 + x + 1) – x^8. (x^2 + x + 1) + x^6. (x^2 + x + 1) – x^5. (x^2 + x + 1) + x^3. (x^2 + x + 1) – x^2. (x^2 + x + 1) + 1. (x^2 + x + 1)
= (x^2 + x + 1). (x^9 – x^8 + x^6 – x^5 + x^3 – x^2 + 1)
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d) x^5 + x^4 + 1
= x^5 + x^4 + x^2 – x^2 + 1
= (x^5 – x^2) + (x^4 + x^2 + 1)
= x^2. (x^3 – 1) + (x^4 + 2x^2 + 1 – x^2)
= x^3. (x – 1). (x^2 + x + 1) + [(x^2 + 1)^2 – x^2]
= x^3. (x – 1). (x^2 + x + 1) + (x^2 + 1 + x). (x^2 + 1 – x)
= (x^2 + x + 1). [x^3. (x – 1) + (x^2 + 1 – x)]
= (x^2 + x + 1). (x^4 – x^3 + x^2 + 1 – x)