Question

Phân tích đa thức thành nhân tử:

\(\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)+15\)

Answer 1

(x+1)(x+3)(x+5)(x+8)+15

=[(x+1)(x+7)][(x+3)(x+5)]+15

=(x²+8x+7)(x²+8x+15)+15

Đặt t=x²+8x+7

=>x²+8x+15=t+8

=>(x² +8x+7)(x²+8x+15)+15

=t(t+8)+15

=t²+8t+15

=t²+3t+5t+15

=t(t+3)+5(t+3)

=(t+3)(t+5)

=(x²+8x+10)(x²+8x+12)

Answer 2

Đặt \(A=\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)+15\)

\(\Rightarrow A=\left(x+1\right)\left(x+7\right)\left(x+3\right)\left(x+5\right)+15\)

        \(=\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\)

Đặt \(x^2+8x+11=t\)

\(\Rightarrow A=\left(t-4\right)\left(t+4\right)+15=t^2-16+15=t^2-1=\left(t+1\right)\left(t-1\right)\)

\(=\left(x^2+8x+11+1\right)\left(x^2+8x+11-1\right)=\left(x^2+8x+12\right)\left(x^2+8x+10\right)\)

\(=\left(x^2+2x+6x+12\right)\left(x^2+8x+10\right)\)\(=\left[x\left(x+2\right)+6\left(x+2\right)\right]\left(x^2+8x+10\right)\)

\(=\left(x+2\right)\left(x+6\right)\left(x^2+8x+10\right)\)