Question

Hello, I need some help with Part 2 question 4! Please show work as the instructions asked!

Answer 1

• Potential Roots are: 1, -1, 1/2, -1/2, 2,-2,4, and -4.

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• Actual roots: -1 and -2.

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• Code piece: H.

Explanation:

Given the polynomial:

`2x^3+8x^2+10x+4`

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:

`\begin{gathered} (p)/(q)=\pm(1)/(1),\pm(1)/(2),\pm(2)/(1),\pm(2)/(2),\pm(4)/(1),\pm(4)/(2) \\ (p)/(q)=\pm1,\pm(1)/(2),\pm2,\pm4 \end{gathered}`

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:

`\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}`

From the calculations above, the actual roots are -1 and -2.

Thus, the actual roots are:

`x=-2;x=-1`

The code piece is H.