if 2 is a zero of a polynomial q(x) then 2 is also a zero of 2×q(x)?
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Considering a cubic polynomial,
p(x)=ax
3
+bx
2
+cx+d
Let α,β,γ be the zeros of the polynomial.
Then,two of these must be zero.
Let α=β=0.
Since,α=0 is a zero of the polynomial,so p(0)=0
=>a(0)
3
+b(0)
2
+c(0)+d=0
=>0+0+0+d=0
=>d=0
Therefore p(x) reduces to
p(x)=ax
3
+bx
2
+cx=x(ax
2
+bx+c)
For the zeros we have p(x)=0
=>x(ax
2
+bx+c)=0
=>x=0or (ax
2
+bx+c)=0 (i)
Also given that β=0 is the zero of polynomial.
From equation (i) we get,
=>a(0)
2
+b(0)+c=0
=>c=0
Hence c=0 and d=0
Therefore p(x) does not have linear and constant term.
Hence, the given statement is true.
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