Given
One of the zeros of 14² − 42² − 9 is the negative of the other.
To find the value of k.
Explanation:
It is given that,
One of the zeros of 14² − 42² − 9 is the negative of the other.
That implies,
Let α, -α be the zeros of the polynomial f(x) = 14² − 42² − 9.
Then,
[tex]\begin{gathered} f(\alpha)=0 \\ \Rightarrow14\alpha^2-42k^2\alpha-9=0\text{ \_\_\_\_\_\_\lparen1\rparen} \end{gathered}[/tex]Also,
[tex]\begin{gathered} f(-\alpha)=0 \\ \Rightarrow14\alpha^2+42k^2\alpha-9=0\text{ \_\_\_\_\_\_\lparen2\rparen} \end{gathered}[/tex]Subtracting (1) and (2) implies,
[tex]\begin{gathered} (2)-(1)\Rightarrow(14\alpha^2+42k^2\alpha-9)-(14\alpha^2-42k^2\alpha-9)=0 \\ \Rightarrow84k^2\alpha=0 \\ \Rightarrow k^2\alpha=0 \\ \because\alpha\ne0\Rightarrow k^2=0 \\ \Rightarrow k=0 \end{gathered}[/tex]Hence, the value of k is 0.
