Phân tích đa thức thành nhân tử : (x2 – 3x)2 – 14x2 + 42x + 40
Phân tích đa thức thành nhân tử : x2 – 3x – 15
x2-2x-15=(x2-5x)+(3x-15)=x(x-5)+3(x-5)=(x-5)(x+3)
\(x^2-3x-15=\left(x^2-2.\dfrac{3}{2}x+\dfrac{9}{4}\right)-\dfrac{69}{4}=\left(x-\dfrac{3}{2}\right)^2-\left(\dfrac{\sqrt{69}}{2}\right)^2\)
\(=\left(x-\dfrac{3}{2}-\dfrac{\sqrt{69}}{2}\right)\left(x-\dfrac{3}{2}+\dfrac{\sqrt{69}}{2}\right)\)
\(x^2-2x-15=\left(x-5\right)\left(x+3\right)\)
Phân tích đa thức sau thành nhân tử : (x2 + 4x + 8)2 + 3x(x2 + 4x + 8) + 2x2
\(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2=\left(x^2+4x+8\right)^2+2x\left(x^2+4x+8\right)+x\left(x^2+4x+8\right)+2x^2\)
\(=\left(x^2+4x+8\right)\left(x^2+4x+8+2x\right)+x\left(x^2+4x+8+2x\right)\)
\(=\left(x^2+4x+8\right)\left(x^2+6x+8\right)+x\left(x^2+6x+8\right)\)
\(=\left(x^2+4x+8+x\right)\left(x^2+6x+8\right)=\left(x^2+5x+8\right)\left(x^2+6x+8\right)\)
Ta có: \(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2\)
\(=\left(x^2+5x+8\right)\left(x^2+6x+8\right)\)
\(=\left(x^2+5x+8\right)\left(x+2\right)\left(x+4\right)\)
Phân tích đa thức thành nhân tử : x2 – 2xy + y2 + 3x – 3y – 10
x2-2xy+y2+3x-3y-10
= (x-y)2+3(x-y)-10
= [(x-y)2+5(x-y)]-[2(x-y)+10]
= (x-y)(x-y+5)-2(x-y+5)
= (x-y+5)(x-y-2)
Ta có: \(x^2-2xy+y^2+3x-3y-10\)
\(=\left(x-y\right)^2+3\left(x-y\right)-10\)
\(=\left(x-y+5\right)\left(x-y-2\right)\)
Phân tích đa thức thành nhân tử : (1 + x2)2 – 4x(1 – x2)
(1 + x2)2 - 4x(1 - x2)
= (1 + x2)(1 + x2) - 4x(1 - x2)
= (1 + x2 - 4x)(1 + x2 - 1 + x2)
= 2x2(x2 - 4x + 1)
Ta có: \(\left(x^2+1\right)^2+4x\left(x^2-1\right)\)
\(=x^4+2x^2+1+4x^3-4x\)
\(=x^4+2x^3+2x^3+4x^2-2x^2-4x+1\)
\(=\left(x+2\right)\left(x^3+2x^2-2x\right)+1\)
Phân tích đa thức thành nhân tử : (x2 + x)2 + 4x2 + 4x – 12
\(\left(x^2+x\right)^2+4x^2+4x-12=\left[\left(x^2+x\right)^2+4\left(x^2+x\right)+4\right]-16=\left(x^2+x+2\right)-4^2=\left(x^2+x+2-4\right)\left(x^2+x+2+4\right)=\left(x^2+x-2\right)\left(x^2+x+6\right)=\left(x-1\right)\left(x+2\right)\left(x^2+x+6\right)\)
\(\left(x^2+x\right)^2+4x^2+4x-12\\ =\left(x^2+x+2\right)-4\\ =\left(x^2+x-2\right)\left(x^2+x+6\right)\)
\(\left(x^2+x\right)^2+4x^2+4x-12\)
\(=\left(x^2+x+6\right)\left(x^2+x-2\right)\)
\(=\left(x^2+x+6\right)\left(x+2\right)\left(x-1\right)\)
Phân tích đa thức thành nhân tử : (x2 – 5x)2 – 3x2 + 15x – 18
\(\left(x^2-5x\right)^2-3x^2+15x-18\)
\(=\left(x^2-5x\right)^2-3\left(x^2-5x\right)-18\)
\(=\left(x^2-5x-6\right)\left(x^2-5x+3\right)\)
\(=\left(x^2-5x+3\right)\left(x-6\right)\left(x+1\right)\)
\(=\left(x^2-5x\right)^2-3\left(x^2-5x\right)-18\\ =\left(x^2-5x\right)^2-6\left(x^2-5x\right)+3\left(x^2-5x\right)-18\\ =\left(x^2-5x\right)\left(x^2-5x-6\right)+3\left(x^2-5x-6\right)\\ =\left(x^2-5x+3\right)\left(x^2-5x-6\right)\\ =\left(x-6\right)\left(x+1\right)\left(x^2-5x+3\right)\)
\(=x^4-10x^3+25x^2-3x^2+15x-18=x^4-10x^3+22x^2+15x-18=x^4+x^3-11x^3-11x^2+33x^2+33x-18x-18=x^3\left(x+1\right)-11x^2\left(x+1\right)+33x\left(x+1\right)-18\left(x+1\right)=\left(x+1\right)\left(x^3-11x^2+33x-18\right)=\left(x+1\right)\left(x^3-6x^2-5x^2+30x+3x-18\right)=\left(x+1\right)\left[x^2\left(x-6\right)-5x\left(x-6\right)+3\left(x-6\right)\right]=\left(x+1\right)\left(x-6\right)\left(x^2-5x\right)=\left(x+1\right)\left(x-6\right)x\left(x-5\right)\)
phân tích đa thức thành nhân tử : 3x^2-7x+4
\(3x^2-7x+4=\left(3x^2-3x\right)-\left(4x-4\right)=3x\left(x-1\right)-4\left(x-1\right)=\left(x-1\right)\left(3x-4\right)\)
3x^2 -7x+4
= 3x^2 -3x-4x+4
= 3x ( x-1) -4(x-1)
= (3x-4)(x-1)
3x^2 - 7x + 4 =3x ^2 - 4x - 3x + 4 = 3x(x-1) - 4 ( x+ 1)= (3x-4)(x-1)
Phân tích đa thức sau thành nhân tử : x2(x + 4)2 – (x + 4)2 – (x2 – 1)
\(x^2\left(x+4\right)^2-\left(x+4\right)^2-\left(x^2-1\right)\\ =\left(x+4\right)^2\left(x^2-1\right)-\left(x^2-1\right)\\ =\left(x^2-1\right)\left[\left(x+4\right)^2-1\right]\\ =\left(x-1\right)\left(x+1\right)\left(x+4-1\right)\left(x+4+1\right)\\ =\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+5\right)\)
\(= (x+4)^2(x^2-1)-(x^2-1)=[(x+4)^2-1](x^2-1)\)
\(=(x+4-1)(x+4+1)(x-1)(x+1)\)
\(=(x+3)(x+5)(x-1)(x+1)\)
\(x^2\left(x+4\right)^2-\left(x+4\right)^2-\left(x^2-1\right)\)
\(=\left(x+4\right)^2\left(x^2-1\right)-\left(x^2-1\right)\)
\(=\left(x^2-1\right)\left[\left(x+4\right)^2-1\right]\)
\(=\left(x^2-1\right)\left(x+3\right)\left(x+5\right)\)
Phân tích đa thức thành nhân tử : x2 - 2x - 24
\(x^2-2x-24\)
\(=x^2-6x+4x-24\)
\(=x(x-6)+4(x-6)\)
\(=(x+4)(x-6)\)
\(x^2-2x-24\\ =x^2-2x+1-25\\ =\left(x-1\right)^2-5^2\\ =\left(x-1-5\right)\left(x-1+5\right)\\ =\left(x-6\right)\left(x+4\right)\)
\(x^2-2x-24=\left(x-6\right)\left(x+4\right)\)