a) \(2a^3+7a^2b+7ab^2+2b^3\)
\(=2\left(a^3+b^3\right)+7ab\left(a+b\right)\)
\(=2\left(a+b\right)\left(a^2-ab+b^2\right)+7ab\left(a+b\right)\)
\(=\left(a+b\right)\left(2a^2-2ab+b^2+7ab\right)\)
\(=\left(a+b\right)\left(2a^2+5ab+2b^2\right)\)
b) \(x^4+2x^2y+y^2-9\)
\(=\left(x^4+2x^2y+y^2\right)-9\)
\(=\left(x^2+y\right)^2-3^2\)
\(=\left(x^2+y-3\right)\left(x^2+y+3\right)\)
c) \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=\left(x^2+7x+11-1\right)\left(x^2+7x+11+1\right)-24\)
\(=\left(x^2+7x+11\right)^2-1-24\)
\(=\left(x^2+7x+11\right)^2-5^2\)
\(=\left(x^2+7x+16\right)\left(x^2+7x+6\right)\)
\(=\left(x^2+7x+16\right)\left(x+1\right)\left(x+6\right)\)
d) \(x^4-6x^2-7x-6\)
\(=x^4-3x^3+3x^3-9x^2+3x^2-9x+2x-6\)
\(=x^3\left(x-3\right)+3x^2\left(x-3\right)+3x\left(x-3\right)+2\left(x-3\right)\)
\(=\left(x^3+3x^2+3x+2\right)\left(x-3\right)\)
\(=\left(x^3+2x^2+x^2+2x+x+2\right)\left(x-3\right)\)
\(=\left[x^2\left(x+2\right)+x\left(x+2\right)+\left(x+2\right)\right]\left(x-3\right)\)
\(=\left(x+2\right)\left(x^2+x+1\right)\left(x-3\right)\)
Câu 1
a) \(P=2a^3+7a^2b+7ab^2+2b^3\)
\(=\left(2a^3+2b^3\right)+\left(7a^2b+7ab^2\right)\)
\(=2\left(a^3+b^3\right)+7ab\left(a+b\right)\)
\(=2\left(a+b\right)\left(a^2-ab+b^2\right)+7ab\left(a+b\right)\)
\(=\left(a+b\right)\left[2\left(a^2-ab+b^2\right)+7ab\right]\)
\(=\left(a+b\right)\left(2a^2-2ab+2b^2+7ab\right)\)
\(=\left(a+b\right)\left(2a^2+5ab+2b^2\right)\)
\(=\left(a+b\right)\left(2a^2+4ab+ab+2b^2\right)\)
\(=\left(a+b\right)\left[\left(2a^2+4ab\right)+\left(ab+2b^2\right)\right]\)
\(=\left(a+b\right)\left[2a\left(a+2b\right)+b\left(a+2b\right)\right]\)
\(=\left(a+b\right)\left(a+2b\right)\left(2a+b\right)\)
b) \(x^4+2x^2y+y^2-9\)
\(=\left(x^2\right)^2+2x^2y+y^2-3^2\)
\(=\left(x^2+y\right)^2-3^2\)
\(=\left(x^2+y-3\right)\left(x^2+y+3\right)\)
c) \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(=\left(x^2+5x+2x+10\right)\left(x^2+4x+3x+12\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\) (1)
Đặt \(t=x^2+7x+10\)
\(\Rightarrow\left(1\right)=t\left(t+2\right)-24\)
\(=t^2+2t-24\)
\(=t^2+2t+1-25\)
\(=\left(t+1\right)^2-5^2\)
\(=\left(t+1-5\right)\left(t+1+5\right)\)
\(=\left(t-4\right)\left(t+6\right)\) (2)
Thế \(t=x^2+7x+10\) vào (2), ta có:
\(\left(2\right)=\left(x^2+7x+10-4\right)\left(x^2+7x+10+6\right)\)
\(=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(=\left(x^2+x+6x+6\right)\left(x^2+7x+16\right)\)
\(=\left[\left(x^2+x\right)+\left(6x+6\right)\right]\left(x^2+7x+16\right)\)
\(=\left[x\left(x+1\right)+6\left(x+1\right)\right]\left(x^2+7x+16\right)\)
\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
d) \(x^4-6x^2-7x-6\)
\(=x^4-3x^3+3x^3-9x^2+3x^2-9x+2x-6\)
\(=\left(x^4-3x^3\right)+\left(3x^3-9x^2\right)+\left(3x^2-9x\right)+\left(2x-6\right)\)
\(=x^3\left(x-3\right)+3x^2\left(x-3\right)+3x\left(x-3\right)+2\left(x-3\right)\)
\(=\left(x-3\right)\left(x^3+3x^2+3x+2\right)\)
\(=\left(x-3\right)\left(x^3+2x^2+x^2+2x+x+2\right)\)
\(=\left(x-3\right)\left[\left(x^3+2x^2\right)+\left(x^2+2x\right)+\left(x+2\right)\right]\)
\(=\left(x-3\right)\left[x^2\left(x+2\right)+x\left(x+2\right)+\left(x+2\right)\right]\)
\(=\left(x-3\right)\left(x+2\right)\left(x^2+x+1\right)\)
e) \(x^4+2008x^2+2007x+2008\)
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