1. Phân tích các đa thức sau đây thành nhân tử.
a) a³ – 7a – 6
b) a³ + 4a² – 7a – 10
c) a(b + c)² +b(c + a )² + c(a +b )² – 4abc
d (a² + a)² + 4(a² + a) – 12
e) (x² + x + 1)( x² + x + 2) -12
g) x⁸ + x + 1
h) x¹⁰ + x⁵ + 1
i) x³ ( z -y² ) + y³ ( x – z² ) + z³ ( y – x² ) + xyz( xyz – 1 )
k) x(y – z)² + y(z – x)² + z(x – y)² – x³ – y³ – z³ + 4xyz
l) (x + y + z)³ – (x + y – z)³ – (y + z – x)³ – (z + x – y)³
1. Phân tích các đa thức sau đây thành nhân tử. a) a³ – 7a – 6 b) a³ + 4a² – 7a – 10 c) a(b + c)² +b(c + a )² + c(a +b )² – 4abc d (a² + a)² + 4
By Nevaeh
Đáp án:
Giải thích các bước giải:
c)a(b+c)²+b(c+a)²+c(a+b)²-4abc
= ab²+bc²+2abc+bc²+ba²+2abc+ca²+cb²+…
=ab²+ac²+bc²+ba²+ca²+cb²+2abc
=ab²+ac²+bc²+ba²+ca²+cb²+2abc
=ab(b+c)+ac(b+c)+cb(b+c)+a²(b+c)
=(b+c)[a(a+c)+b(a+c)]
=(a+b)(a+c)(b+c)
g)x⁸ + x + 1
= x⁸-x²+x²+x+1
= x²($x^{6}$ -1)+x²+x+1
= x²(x³+1)(x³-1)+ x²+x+1
=(x²+x+1)(x²(x³+1)(x-1)+1)
a) a³ – 7a – 6
= a³ + a² – a² – a – 6a – 6
= a²(a + 1) – a(a + 1) – 6(a + 1)
= (a + 1)(a² – a – 6)
= (a + 1)(a² – 3a + 2a – 6)
= (a + 1)[a(a – 3) + 2(a – 3)]
= (a + 1)(a – 3)(a + 2)
b) a³ + 4a² – 7a – 10
= a³ – 2a² + 6a² – 12a + 5a – 10
= a²(a – 2) + 6a(a – 2) + 5(a – 2)
= (a – 2)(a² + 6a + 5)
= (a – 2)(a² + a + 5a + 5)
= (a – 2)[a(a + 1) + 5(a + 1)]
= (a – 2)(a + 1)(a + 5)
c) a(b + c)² + b(c + a )² + c(a + b )² – 4abc
= ab² + bc² + 2abc + bc² + ba² + 2abc + ca² + cb² + …
= ab² + ac² + bc² + ba² + ca² + cb² + 2abc
= ab(b + c) + ac(b + c) + cb(b + c) + a²(b + c)
= (b + c)[a(a + c) + b(a + c)]
= (a + b)(a + c)(b + c)
e) (x² + x + 1)( x² + x + 2) – 12
= (x² + x + 1)[(x² + x + 1) + 1] – 12
= (x² + x + 1)² + (x² + x + 1) – 12
= (x² + x + 1)² – 3(x² + x + 1) + 4(x² + x + 1) – 12
= (x² + x + 1)(x² + x + 1 – 3) + 4(x² + x + 1 – 3)
= (x² + x + 1)(x² + x – 2) + 4(x² + x – 2)
= (x² + x + 2)(x² + x + 1 + 4)
= (x² – x + 2x – 2)(x² + x + 5)
= [x(x – 1) + 2(x – 1)](x² + x + 5)
= (x – 1)(x + 2)(x² + x + 5)
g) x⁸ + x + 1
= $x^{8}$ + $x^{7}$ – $x^{7}$ + $x^{6}$ – $x^{6}$ + $x^{5}$ – $x^{5}$ + $x^{4}$ – $x^{4}$ + $x^{3}$ – $x^{3}$ + $x^{2}$ – $x^{2}$ + 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) + ($x^{2}$ + x + 1)
= ($x^{2}$ + x + 1)($x^{6}$ – $x^{5}$ + $x^{3}$ – $x^{2}$ + 1)