# Check whether p(x) is a multiple of g(x) or not p(x)=x^3-5x^2+4x-3, g(x)=x-2

Check whether p(x) is a multiple of g(x) or not
$p\left(x\right)={x}^{3}-5{x}^{2}+4x-3$
$g\left(x\right)=x-2$
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Elisa Spears
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Given
$p\left(x\right)={x}^{3}-5{x}^{2}+4x-3$
$g\left(x\right)=x-2Step 2 Finding the given$
According to the question,
$=g\left(x\right)=x-2$
$g\left(x\right)=0$
$x-2=0$
$x=2$
zero of $g\left(x\right)=2$
So, substituning the value of x in p(x).
$p\left(2\right)=\left(2{\right)}^{3}-5\left(2{\right)}^{2}+4\left(2\right)-3$
$p\left(2\right)=\left(2{\right)}^{3}-5\left(2{\right)}^{2}+4\left(2\right)-3$
$=-7\ne 0$
\therefore p(x) is not the multiple of g(x) since the remainder $\ne 0$
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Inbrunstlr
Given, $p\left(x\right)={x}^{3}-5{x}^{2}+4x-3$
$g\left(x\right)=x-2$
$⇒x-2=0$
$⇒x=2$
Now, p(2) should be 0 if p(x) is a multiple of g(x)
Thus,
$p\left(2\right)={2}^{3}-5\left(2{\right)}^{2}+4\left(2\right)-3$
$=8-20+8-3$
$=-7$
$=8-20+8-3$
$=0$
Thus, p(x) is not a multiple of g(x)