Question

$show \: that \\ find \: the \: value \: of \: a \: and \: b \: of \: \\ (x - 1) \: and \: (x - 2) \: will \: be \: \\ factors \: of \: f \: (x) = {x}^3 \: - {10x}^2 \\ + a \: x + b$
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Answer 1

Given:-

$\text{(x-1) and (x-2) are the factors of }x^3-10x^2+ax+b.$

To find:-

$\text{The values of `a' and `b'.}$

Solution:-

$\text{Zero of (x-1) and (x-2):}\\(x-1)=0\\\implies x=1\\\\(x-2)=0\\\implies x=2$

$\text{Now,}$

$(1)^3-10(1)^2+a(1)+b=(2)^3-10(2)^2+a(2)+b=0\\\\\implies 1-10+a+b=8-40+2a+b\\\\\implies -9+a+b=-32+2a+b\\\\\implies 23+a+b=2a+b\\\\\boxed{\implies 23=a}$

$\text{Now, putting the value of a=23 in the equation:}$

$1-10+a+b=0\\\\\implies -9+23+b=0\\\\\implies 14+b=0\\\\\boxed{\implies b=-14}$