To determine between which consecutive integers the real zeros of the polynomial [tex]\( f(x) = 3x^3 - 5x^2 + 5x + 7 \)[/tex] are located on the interval [tex]\([-10,10]\)[/tex], we need to find the roots of the polynomial first and then check where these roots lie within the given intervals.
From our previous steps, the roots of the polynomial [tex]\( f(x) = 3x^3 - 5x^2 + 5x + 7 \)[/tex] are calculated to be:
[tex]\[ 1.18397262 + 1.38761118i \][/tex]
[tex]\[ 1.18397262 - 1.38761118i \][/tex]
[tex]\[ -0.70127858 \][/tex]
To determine between which consecutive integers the real zeros are located, we focus on the real part of the roots, neglecting the imaginary components. The only real root we have here is:
[tex]\[ -0.70127858 \][/tex]
We now place this root in the appropriate interval:
- [tex]\(-0.70127858\)[/tex] lies between [tex]\(-1\)[/tex] and [tex]\(0\)[/tex].
Therefore, the real zero of [tex]\( f(x) \)[/tex] is located between the integers [tex]\(-1\)[/tex] and [tex]\(0\)[/tex].
So, the intervals which contain the real zeros are:
[tex]\[ -1 < x < 0.\][/tex]
Here are the relevant intervals from the given choices:
[tex]\[ -1 < x < 0 \][/tex]
[tex]\[ 0 < x < 1 \][/tex]
[tex]\[ 1 < x < 2 \][/tex]
[tex]\[ 2 < x < 3 \][/tex]
[tex]\[ -1 < x < 0 \][/tex]
[tex]\[ 1 < x < 2 \][/tex]
[tex]\[ -8 < x < -7 \][/tex]
Based on the calculation:
- The interval containing the root is: [tex]\(-1 < x < 0 \)[/tex].
So, the final answer for the real zeros of the polynomial [tex]\( f(x)=3 x^3-5 x^2+5 x+7 \)[/tex] is:
[tex]\[ -1 < x < 0 \][/tex]
