Use synthetic division to determine the multiplicity of the root.

The value x=-2 is a zero of the function g(x)=x^(4)+3x^(3)-4x. Use synthetic division to determine the multiplicity of the root. The value x=-2 is a zero of the function g(x)-x^(4)+3x^(3)-4x+1 the multiplicity of the root.

The Correct Answer and Explanation is:

To determine the multiplicity of the root x=−2x = -2x=−2 for the function g(x)=x4+3×3−4x+1g(x) = x^4 + 3x^3 – 4x + 1g(x)=x4+3×3−4x+1, we will use synthetic division.

Step 1: Set up the synthetic division.

First, we write the coefficients of the polynomial g(x)=x4+3×3+0x2−4x+1g(x) = x^4 + 3x^3 + 0x^2 – 4x + 1g(x)=x4+3×3+0x2−4x+1. The coefficient of x2x^2×2 is zero because there is no x2x^2×2 term in the polynomial.

The coefficients are:
1, 3, 0, -4, 1.

We are dividing by x+2x + 2x+2 because x=−2x = -2x=−2 is a root, so the divisor is −2-2−2.

Step 2: Perform synthetic division.

We use the synthetic division method to divide the polynomial by x+2x + 2x+2:−2130−41−2−24011−201\begin{array}{r|rrrrr} -2 & 1 & 3 & 0 & -4 & 1 \\ & & -2 & -2 & 4 & 0 \\ \hline & 1 & 1 & -2 & 0 & 1 \\ \end{array}−2​11​3−21​0−2−2​−440​101​​

• Bring down the first coefficient (1).
• Multiply 111 by −2-2−2 and place it under the next coefficient: 1×(−2)=−21 \times (-2) = -21×(−2)=−2.
• Add 3+(−2)=13 + (-2) = 13+(−2)=1.
• Multiply 111 by −2-2−2 and place it under the next coefficient: 1×(−2)=−21 \times (-2) = -21×(−2)=−2.
• Add 0+(−2)=−20 + (-2) = -20+(−2)=−2.
• Multiply −2-2−2 by −2-2−2 and place it under the next coefficient: −2×(−2)=4-2 \times (-2) = 4−2×(−2)=4.
• Add −4+4=0-4 + 4 = 0−4+4=0.
• Multiply 000 by −2-2−2 and place it under the next coefficient: 0×(−2)=00 \times (-2) = 00×(−2)=0.
• Add 1+0=11 + 0 = 11+0=1.

The result of the synthetic division is:x3+x2−2x+0+1x+2x^3 + x^2 – 2x + 0 + \frac{1}{x + 2}x3+x2−2x+0+x+21​

Step 3: Interpretation of the result.

Since the remainder is 1, x=−2x = -2x=−2 is not a factor of g(x)g(x)g(x), meaning it is not a root of the function. Thus, the root x=−2x = -2x=−2 does not have multiplicity in this function.

Conclusion:

The value x=−2x = -2x=−2 is not a zero of g(x)=x4+3×3−4x+1g(x) = x^4 + 3x^3 – 4x + 1g(x)=x4+3×3−4x+1, and there is no multiplicity associated with it.++-=