Answer:
• Potential Roots are: 1, -1, 1/2, -1/2, 2,-2,4, and -4.
,
• Actual roots: -1 and -2.
,
• Code piece: H.
Explanation:
Given the polynomial:
[tex]2x^3+8x^2+10x+4[/tex]
Applying the rational Toot theorem:
The constant = 4
• The factors of the constant, p = ±1,±2, and ±4
The leading coefficient = 2
• The factors of the leading coefficient, q = ±1 and ±2.
The potential roots are obtained below:
[tex]\begin{gathered} \frac{p}{q}=\pm\frac{1}{1},\pm\frac{1}{2},\pm\frac{2}{1},\pm\frac{2}{2},\pm\frac{4}{1},\pm\frac{4}{2} \\ \frac{p}{q}=\pm1,\pm\frac{1}{2},\pm2,\pm4 \end{gathered}[/tex]
Potential Roots are: 1, -1, 1/2, -1/2, 2,-2,4, and -4.
Next, find the actual roots by substituting each of the potential roots for x:
[tex]\begin{gathered} f(1)=2(1)^3+8(1)^2+10(1)+4=24 \\ f(-1)=2(-1)^3+8(-1)^2+10(-1)+4=0 \\ f(0.5)=2(0.5)^3+8(0.5)^2+1(0.5)+4=11.25 \\ f(-0.5)=2(-0.5)^3+8(-0.5)^2+1(-0.5)+4=0.75 \\ f(2)=2(2)^3+8(2)^2+10(2)+4=72 \\ f(-2)=2(-2)^3+8(-2)^2+10(-2)+4=0 \\ f(4)=2(4)^3+8(4)^2+10(4)+4=300 \\ f(-4)=2(-4)^3+8(-4)^2+10(-4)+4=-36 \end{gathered}[/tex]
From the calculations above, the actual roots are -1 and -2.
Thus, the actual roots are:
[tex]x=-2;x=-1[/tex]
The code piece is H.
