Answer:
One zero of the polynomial ax^2+bx+cax
2
+bx+c is reciprocal of the other.
Assume that one of the zero of above polynomial as x, then another zero will be 1/x.
Product of zeroes
\begin{gathered}\begin{aligned} = ( x ) \left( \frac { 1 } { x } \right) & = \frac { \text { constant } } { x ^ { 2 } \text { coefficient } } \\\\ 1 & = \frac { c } { a } \\\\ a & = c \end{aligned}\end{gathered}
=(x)(
x
1
)
a
=
coefficient
constant
c
=c
Let us take one polynomial to find that when a = c, zeros are reciprocal.
\begin{gathered}\begin{array} { l } { 4 \mathrm { x } ^ { 2 } + 10 \mathrm { x } + 4 = 0 } \\\\ { 4 \mathrm { x } ^ { 2 } + 8 \mathrm { x } + 2 \mathrm { x } + 4 = 0 } \\\\ { 4 \mathrm { x } ( \mathrm { x } + 2 ) + 2 ( \mathrm { x } + 2 ) = 0 } \\\\ { ( \mathrm { x } + 2 ) ( 4 \mathrm { x } + 2 ) = 0 } \end{array}\end{gathered}
4x
+10x+4=0
+8x+2x+4=0
4x(x+2)+2(x+2)=0
(x+2)(4x+2)=0
First zero = x + 2 i.e. x = -2
Second zero = 4x + 2 i.e. 4x = -2 then x = -1/2
Hence, it can be said that a = c, then zeros are reciprocal.
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Answers & Comments
Answer:
One zero of the polynomial ax^2+bx+cax
2
+bx+c is reciprocal of the other.
Assume that one of the zero of above polynomial as x, then another zero will be 1/x.
Product of zeroes
\begin{gathered}\begin{aligned} = ( x ) \left( \frac { 1 } { x } \right) & = \frac { \text { constant } } { x ^ { 2 } \text { coefficient } } \\\\ 1 & = \frac { c } { a } \\\\ a & = c \end{aligned}\end{gathered}
=(x)(
x
1
)
1
a
=
x
2
coefficient
constant
=
a
c
=c
Let us take one polynomial to find that when a = c, zeros are reciprocal.
\begin{gathered}\begin{array} { l } { 4 \mathrm { x } ^ { 2 } + 10 \mathrm { x } + 4 = 0 } \\\\ { 4 \mathrm { x } ^ { 2 } + 8 \mathrm { x } + 2 \mathrm { x } + 4 = 0 } \\\\ { 4 \mathrm { x } ( \mathrm { x } + 2 ) + 2 ( \mathrm { x } + 2 ) = 0 } \\\\ { ( \mathrm { x } + 2 ) ( 4 \mathrm { x } + 2 ) = 0 } \end{array}\end{gathered}
4x
2
+10x+4=0
4x
2
+8x+2x+4=0
4x(x+2)+2(x+2)=0
(x+2)(4x+2)=0
First zero = x + 2 i.e. x = -2
Second zero = 4x + 2 i.e. 4x = -2 then x = -1/2
Hence, it can be said that a = c, then zeros are reciprocal.