Factorization involves splitting a function into factors..
The factorized expression is [tex]P(x)= (x - 3)(x - 2) (x -1)[/tex]
The function is given as:
[tex]p(x) =x^3 - 6x^2 + 11x - 6[/tex]
Start by calculating p(1) to determine if x - 1 is a root of the polynomial
[tex]p(1) =1^3 - 6(1)^2 + 11(1) - 6[/tex]
[tex]p(1) =0[/tex]
Since P(1) equals 0, then x - 1 is a root of the polynomial.
Divide both sides of [tex]p(x) =x^3 - 6x^2 + 11x - 6[/tex] by x - 1
[tex]\frac{P(x)}{x - 1} = \frac{x^3 - 6x^2+11x - 6}{x - 1}[/tex]
Factorize the numerator
[tex]\frac{P(x)}{x - 1} = \frac{( x - 1 )(x^2 - 5x + 6 )}{x - 1}[/tex]
Cancel out common factor
[tex]\frac{P(x)}{x - 1} = x^2 - 5x + 6[/tex]
Expand
[tex]\frac{P(x)}{x - 1} = x^2 - 2x - 3x + 6[/tex]
Factorize
[tex]\frac{P(x)}{x - 1} = x(x - 2) - 3(x - 2)[/tex]
Factor out x - 2
[tex]\frac{P(x)}{x - 1} = (x - 3)(x - 2)[/tex]
Multiply though by x - 1
[tex]P(x)= (x - 3)(x - 2) (x -1)[/tex]
Hence, the factorized expression is [tex]P(x)= (x - 3)(x - 2) (x -1)[/tex]
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