To find the other zeros of the polynomial [tex]\( f(x) = x^3 + 7x^2 - 2x - 14 \)[/tex] given that one of the zeros is [tex]\(-7\)[/tex], we can proceed through the following steps:
1. Given Zero: We know that [tex]\( x = -7 \)[/tex] is a zero of the polynomial. This means that [tex]\( f(-7) = 0 \)[/tex].
2. Polynomial Division: Since [tex]\(-7\)[/tex] is a zero, the polynomial [tex]\( f(x) \)[/tex] can be factored as [tex]\( (x + 7) \)[/tex] times another polynomial. We need to divide [tex]\( f(x) \)[/tex] by [tex]\( (x + 7) \)[/tex] to find this other polynomial (the quotient).
3. Quotient Polynomial: On dividing [tex]\( f(x) \)[/tex] by [tex]\( (x + 7) \)[/tex], the quotient polynomial results are given as [tex]\( x^2 - 2 \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] can be factored as:
[tex]\[
f(x) = (x + 7)(x^2 - 2)
\][/tex]
4. Finding Remaining Zeros: The next step is to find the zeros of the quotient polynomial [tex]\( x^2 - 2 \)[/tex]. Setting [tex]\( x^2 - 2 = 0 \)[/tex], we solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 2 = 0
\][/tex]
[tex]\[
x^2 = 2
\][/tex]
[tex]\[
x = \pm\sqrt{2}
\][/tex]
Hence, the solutions to [tex]\( x^2 - 2 = 0 \)[/tex] are [tex]\( x = \sqrt{2} \)[/tex] and [tex]\( x = -\sqrt{2} \)[/tex].
5. Conclusion: In summary, the zeros of the polynomial [tex]\( f(x) = x^3 + 7x^2 - 2x - 14 \)[/tex] are [tex]\( x = -7 \)[/tex], [tex]\( x = \sqrt{2} \)[/tex], and [tex]\( x = -\sqrt{2} \)[/tex].
Therefore, the other zeros of the polynomial are [tex]\( \sqrt{2} \)[/tex] and [tex]\( -\sqrt{2} \)[/tex].
