Possible Using synthetic division, determine whether the numbers are zeros of the polynomial function. \( -1,1 ; \mathrm{h}(\mathrm{x})=\mathrm{x}^{4}+4 \mathrm{x}^{3}+3 \mathrm{x}^{2}-4 \mathrm{x}-4 \)

To determine whether the numbers \(-1\) and \(1\) are zeros of the polynomial function \(h(x) = x^4 + 4x^3 + 3x^2 - 4x - 4\), we will use synthetic division. ### Step 1: Synthetic Division with \(-1\) 1. Write down the coefficients of the polynomial: \(1, 4, 3, -4, -4\). 2. Set up synthetic division with \(-1\): \[ \begin{array}{r|rrrrr} -1 & 1 & 4 & 3 & -4 & -4 \\ & & -1 & -3 & 0 & 4 \\ \hline & 1 & 3 & 0 & -4 & 0 \\ \end{array} \] ### Step 2: Calculate the Result - Bring down the leading coefficient \(1\). - Multiply \(-1\) by \(1\) and add to \(4\): \(4 - 1 = 3\). - Multiply \(-1\) by \(3\) and add to \(3\): \(3 - 3 = 0\). - Multiply \(-1\) by \(0\) and add to \(-4\): \(-4 + 0 = -4\). - Multiply \(-1\) by \(-4\) and add to \(-4\): \(-4 + 4 = 0\). The remainder is \(0\), which means \(-1\) is a zero of the polynomial. ### Step 3: Synthetic Division with \(1\) 1. Set up synthetic division with \(1\): \[ \begin{array}{r|rrrrr} 1 & 1 & 4 & 3 & -4 & -4 \\ & & 1 & 5 & 8 & 4 \\ \hline & 1 & 5 & 8 & 4 & 0 \\ \end{array} \] ### Step 4: Calculate the Result - Bring down the leading coefficient \(1\). - Multiply \(1\) by \(1\) and add to \(4\): \(4 + 1 = 5\). - Multiply \(1\) by \(5\) and add to \(3\): \(3 + 5 = 8\). - Multiply \(1\) by \(8\) and add to \(-4\): \(-4 + 8 = 4\). - Multiply \(1\) by \(4\) and add to \(-4\): \(-4 + 4 = 0\). The remainder is \(0\), which means \(1\) is also a zero of the polynomial. ### Conclusion Both \(-1\) and \(1\) are zeros of the polynomial function \(h(x) = x^4 + 4x^3 + 3x^2 - 4x - 4\).