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\(\text{a, Ta có :}\) \(M=\left(x^2+10x+16\right)\left(x^2+10x+24\right)+16\)
\(\text{Đặt }a=x^2+10x+16\)
\(\text{Ta có: }M=a\left(a+8\right)+16=a^2+8a+16=\left(a+4\right)^2\)
\(M=\left(x^2+10x+20\right)^2\)
\(\text{b, }\)\(\left|x+1\right|=\left|x\left(x+1\right)\right|\)
\(\Leftrightarrow\left|x\left(x+1\right)\right|-\left|x+1\right|=0\)
\(\Leftrightarrow\left|x\right|.\left|x+1\right|-\left|x+1\right|=0\)
\(\Rightarrow\left|x+1\right|\left(\left|x\right|-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left|x+1\right|=0\\\left|x\right|-1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=1\end{matrix}\right.\)
Câu 8 :
\(N=\left(\frac{x-1}{\left(x-1\right)^2+x}-\frac{2}{x-2}\right):\left(\frac{\left(x-1\right)^4+2}{\left(x-1\right)^3-1}-x+1\right)\)
Đặt \(x-1=a\)
\(N=\left(\frac{a}{a^2+x}-\frac{2}{a-1}\right):\left(\frac{a^4+2}{a^3-1}-a\right)\)
\(N=\frac{a\left(a-1\right)-2\left(a^2+x\right)}{\left(a^2+x\right)\left(a-1\right)}:\frac{a^4+2-a\left(a^3-1\right)}{a^3-1}\)
\(N=\frac{a^2-a-2a^2-2x}{\left(a^2+x\right)\left(a-1\right)}:\frac{a^4+2-a^4+a}{a^3-1}\)
\(N=\frac{-a^2-a-2x}{\left(a^2+x\right)\left(a-1\right)}\cdot\frac{\left(a-1\right)\left(a^2+a+1\right)}{2+a}\)
\(N=\frac{-\left(a^2+a+2x\right)\left(a^2+a+1\right)}{\left(a^2+x\right)\left(2+a\right)}\)
\(N=\frac{-\left[\left(x-1\right)^2+x-1+2x\right]\left[\left(x-1\right)^2+x-1+1\right]}{\left[\left(x-1\right)^2+x\right]\left(2+x-1\right)}\)
\(N=\frac{-\left(x^2+x\right)\left(x^2-x+1\right)}{\left(x^2-x+1\right)\left(x+1\right)}\)
\(N=\frac{-x\left(x+1\right)}{x+1}\)
\(N=-x\)( đpcm )
Câu 9 : Tìm giá trị nhỏ nhất của biểu thức :
\(P=\frac{x^2}{x+4}\cdot\left(\frac{x^2+16}{x}+8\right)+9\)
Bài làm :
\(P=\frac{x^2}{x+4}\cdot\frac{x^2+8x+16}{x}+9\)
\(P=\frac{x^2\left(x+4\right)^2}{x\left(x+4\right)}+9\)
\(P=x\left(x+4\right)+9\)
\(P=x^2+4x+9\)
\(P=\left(x+2\right)^2+5\ge5\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x=-2\)
Chắc là \(q\left(x\right)=x^2-4????\)
\(f\left(2\right)=2^5+2^2+1=37\) ; \(f\left(-2\right)=-27\)
Do \(f\left(x\right)\) có 5 nghiệm nên f(x) có dạng:
\(f\left(x\right)=\left(x-x_1\right)\left(x-x_2\right)\left(x-x_3\right)\left(x-x_4\right)\left(x-x_5\right)\)
\(\Rightarrow f\left(2\right)=\left(2-x_1\right)\left(2-x_2\right)\left(2-x_3\right)\left(2-x_4\right)\left(2-x_5\right)=37\)
\(f\left(-2\right)=\left(-2-x_1\right)\left(-2-x_2\right)\left(-2-x_3\right)\left(-2-x_4\right)\left(-2-x_5\right)=-27\)
\(\Rightarrow\left(2+x_1\right)\left(2+x_2\right)\left(2+x_3\right)\left(2+x_4\right)\left(2+x_5\right)=27\)
\(A=\left(x_1^2-4\right)\left(x^2_2-4\right)\left(x_3^2-4\right)\left(x_4^2-4\right)\left(x^2_5-4\right)\)
\(A=-\left(2-x_1\right)\left(2-x_2\right)\left(2-x_3\right)\left(2-x_4\right)\left(2-x_5\right)\left(2+x_1\right)\left(2+x_2\right)\left(2+x_3\right)\left(2+x_4\right)\left(2+x_5\right)\)
\(A=-37.27=-999\)
a)\(P=x^3+6x^2+12x+8+x^3-6x^2+12x-8-2x^3-24x=0\)
Vậy g/t P không phụ thuộc vào biến.
b)\(Q=x^3-3x^2+3x-1-\left(x^3+3x^2+3x+1\right)+6\left(x^2-1\right)=-6x^2-2+6x^2-6=-8\)
Vậy g/t Q không phụ thuộc vào biến.
b) Ta có: \(Q=\left(x-1\right)^3-\left(x+1\right)^3+6\left(x+1\right)\left(x-1\right)\)
\(=\left(x-1-x-1\right)\left[\left(x-1\right)^2+\left(x-1\right)\left(x+1\right)+\left(x+1\right)^2\right]+6\left(x^2-1\right)\)
\(=-2\left(x^2-2x+1+x^2-1+x^2+2x+1\right)+6\left(x^2-1\right)\)
\(=-2\left(3x^2+1\right)+6\left(x^2-1\right)\)
\(=-6x^2-2+6x^2-6\)
=-8
\(M=\left(x+2\right)\left(x+4\right)\left(x+6\right)\left(x+8\right)+16\)
\(M=\left(x^2+10+16\right)\left(x^2+10x+24\right)+16\)
\(M=\left(x^2+16+10x\right)\left(x^6+10x+16+8\right)+16\)
\(M=\left(x^2+10x+16\right)^2+8\left(x^2+10x+16\right)+16\)
\(M=\left(x^2+10x+20\right)^2\left(đpcm\right)\)
