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\(4x^4-8x^3+3x^2-8x+4\)
\(=\left(4x^4-8x^3\right)+\left(3x^2-6x\right)-\left(2x-4\right)\)
\(=4x^3\left(x-2\right)+3x\left(x-2\right)-2\left(x-2\right)\)
\(=\left(x-2\right)\left(4x^3+3x-2\right)\)
x3 - 2x2 - 8x
= x( x2 - 2x - 8 )
= x( x2 - 4x + 2x - 8 )
= x[ x( x - 4 ) + 2( x - 4 ) ]
= x( x - 4 )( x + 2 )
\(x^3-2x^2-8x=x\left(x^2-2x-8\right)=x\left(x^2-2x+1-9\right)=x\left[\left(x-1\right)^2-3^2\right]=x\left(x-4\right)\left(x+2\right)\)
a) x2 - 7x + 5 = ( x2 - 2 . 7/2 . x + 49 / 4 ) + 5 - 49 / 4
= (x - 7/2)^2 - 29/4
= (x - 7/2)^2 - (√ 29 / 2 )^2
= ( x - ( 7 + √ 29 / 2 )). ( x + ( 7 - √ 29 / 2 ))
\(2x^3-5x^2+8x-3\)
\(\Leftrightarrow2x^3-x^2-4x^2+2x+6x+3\)
\(\Leftrightarrow x^2\cdot\left(2x-1\right)-2x\cdot\left(2x-1\right)+3\cdot\left(2x+1\right)\)
\(\Leftrightarrow\left(2x-1\right)\cdot\left(x^2-2x+3\right)\)
\(2x^3-5x^2+8x-3\)
\(=2x^3-x^2-4x^2+2x+6x-3\)
\(=x^2\left(2x-1\right)-2x\left(2x-1\right)+3\left(2x-1\right)\)
\(=\left(2x-1\right)\left(x^2-2x+3\right)\)
= 2x^3 - 4x^2 - x^2 + 2x + 6x - 3
= 2x^2 ( x - 1/2 ) - x ( x - 1/2 ) +3 ( x - 1/2 )
= ( x - 1/2 )( 2x^2 - x + 3 )
\(8x^2-2x-3=8x^2+4x-6x-3=4x\left(2x+1\right)-3\left(2x+1\right)=\left(4x-3\right)\left(2x+1\right)\)
\(8x^2-2x-3=\left(4x-3\right)\left(2x+1\right)\)