Giúp em với huhu
Đặt biến phụ dạng đẳng cấp \( ax^2 + bxy + cy^2 \).
Bài 1. Phân tích các đa thức sau thành nhân tử:
a) \((x^2 + 1)^2 + 3x(x^2 + 1) + 2x^2\) \hspace{1cm} \(t = x^2 + 1\)
b) \((x^2 + 4x + 8)^2 + 3x(x^2 + 4x + 8) + 2x^2\)
c) \(4(x^2 + x + 1)^2 + 5x(x^2 + x + 1) + x^2\) \hspace{1cm} \(t = x^2 + x + 1\)
d) \((x^2 - x + 2)^4 - 3x^2(x^2 - x + 2) + 2x^4\) \hspace{1cm} \(t = (x^2 - x + 2)\)
e) \((x^2 - x - 1)^4 + 7x^2(x^2 - x - 1)^2 + 12x^4\) \hspace{1cm} \(t = (x^2 - x + 1)^2\)
f) \(10(x^2 - 2x + 3)^4 - 9x^2(x^2 - 2x + 3)^2 - x^4\) \hspace{1cm} \(t = (x^2 - 2x + 3)^2\)
a: \(\left(x^2+1\right)^2+3x\left(x^2+1\right)+2x^2\)
\(=\left(x^2+1\right)^2+x\left(x^2+1\right)+2x\left(x^2+1\right)+2x^2\)
\(=\left(x^2+1\right)\left(x^2+x+1\right)+2x\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^2+2x+1\right)=\left(x^2+x+1\right)\left(x+1\right)^2\)
b: \(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2\)
\(=\left(x^2+4x+8\right)^2+x\left(x^2+4x+8\right)+2x\left(x^2+4x+8\right)+2x^2\)
\(=\left(x^2+4x+8\right)\left(x^2+4x+8+x\right)+2x\left(x^2+4x+8+x\right)\)
\(=\left(x^2+5x+8\right)\left(x^2+6x+8\right)\)
\(=\left(x^2+5x+8\right)\left(x^2+2x+4x+8\right)\)
\(=\left(x^2+5x+8\right)\left(x+2\right)\left(x+4\right)\)
c: ,\(4\left(x^2+x+1\right)^2+5x\left(x^2+x+1\right)+x^2\)
\(=4\left(x^2+x+1\right)^2+4x\left(x^2+x+1\right)+x\left(x^2+x+1\right)+x^2\)
\(=4\left(x^2+x+1\right)\left(x^2+x+1+x\right)+x\left(x^2+x+1+x\right)\)
\(=4\left(x^2+x+1\right)\left(x^2+2x+1\right)+x\left(x^2+2x+1\right)\)
\(=\left(x^2+2x+1\right)\left\lbrack4\left(x^2+x+1\right)+x\right\rbrack\)
\(=\left(x+1\right)^2\cdot\left(4x^2+5x+4\right)\)
d: \(\left(x^2-x+2\right)^4-3x^2\left(x^2-x+2\right)^2+2x^4\)
\(=\left(x^2-x+2\right)^4-x^2\cdot\left(x^2-x+2\right)^2-2x^2\left(x^2-x+2\right)^2+2x^4\)
\(=\left(x^2-x+2\right)^2\cdot\left\lbrack\left(x^2-x+2\right)^2-x^2\right\rbrack-2x^2\left\lbrack\left(x^2-x+2\right)^2-x^2\right\rbrack\)
\(=\left\lbrack\left(x^2-x+2\right)^2-x^2\right\rbrack\left\lbrack\left(x^2-x+2\right)^2-2x^2\right\rbrack\)
\(=\left(x^2-x+2-x\right)\left(x^2-x+2+x\right)\left\lbrack x^4+x^2+4-2x^3+4x^2-4x-2x^2\right\rbrack\)
\(=\left(x^2-2x+2\right)\left(-x+2\right)\left(x^4-2x^3+3x^2-4x+4\right)\)
e: \(\left(x^2-x-1\right)^4+7x^2\left(x^2-x-1\right)^2+12x^4\)
\(=\left(x^2-x-1\right)^4+3x^2\left(x^2-x-1\right)^{^2}+4x^2\left(x^2-x-1\right)^2+12x^4\)
\(=\left(x^2-x-1\right)^2\left\lbrack\left(x^2-x-1\right)^2+3x^2\right\rbrack+4x^2\left\lbrack\left(x^2-x-1\right)^2+3x^2\right\rbrack\)
\(=\left\lbrack\left(x^2-x-1\right)^2+3x^2\right\rbrack\left\lbrack\left(x^2-x-1\right)^2+4x^2\right\rbrack\)
f: \(10\left(x^2-2x+3\right)^4-9x^2\left(x^2-2x+3\right)^2-x^4\)
\(=10\left(x^2-2x+3\right)^4-10x^2\left(x^2-2x+3\right)^2+x^2\left(x^2-2x+3\right)^2-x^4\)
\(=10\left(x^2-2x+3\right)^2\left\lbrack\left(x^2-2x+3\right)^2-x^2\right\rbrack+x^2\left\lbrack\left(x^2-2x+3\right)^2-x^2\right\rbrack\)
\(=\left\lbrack\left(x^2-2x+3\right)^2-x^2\right\rbrack\left\lbrack10\left(x^2-2x+3\right)^2+x^2\right\rbrack\)
\(=\left(x^2-2x+3-x\right)\left(x^2-2x+3+x\right)\left\lbrack10\left(x^2-2x+3\right)^2+x^2\right\rbrack\)
\(=\left(x^2-3x+3\right)\left(x^2-x+3\right)\left\lbrack10\left(x^2-2x+3\right)^2+x^2\right\rbrack\)
