For \#'s \( 6-10 \) use vnthetic aivision (1) \( \left(x^{4}-10 x^{2}+27 x^{2}-46 x+28\right) \div(x-7) \)

Certainly! Let's perform the division of the polynomial \[ P(x) = x^{4} - 10x^{2} + 27x^{2} - 46x + 28 \] by \[ D(x) = x - 7. \] First, simplify the polynomial \( P(x) \): \[ P(x) = x^{4} + ( -10x^{2} + 27x^{2} ) - 46x + 28 = x^{4} + 17x^{2} - 46x + 28. \] Now, we'll perform the polynomial long division. --- ### Polynomial Long Division **Step 1:** Set up the division. \[ \begin{array}{r|llll} x - 7 & x^{4} & + 0x^{3} & + 17x^{2} & -46x & +28 \\ \hline \end{array} \] *(Note: We've included the \( 0x^{3} \) term to account for the missing \( x^{3} \) term in \( P(x) \).)* **Step 2:** Divide the highest degree term of the dividend by the highest degree term of the divisor. \[ \frac{x^{4}}{x} = x^{3} \] **Step 3:** Multiply the entire divisor by this result and subtract from the dividend. \[ (x - 7) \times x^{3} = x^{4} - 7x^{3} \] Subtracting from the dividend: \[ \begin{array}{r|llll} x - 7 & x^{4} & + 0x^{3} & + 17x^{2} & -46x & +28 \\ \hline & x^{4} & -7x^{3} & & & \\ \hline & 0 & +7x^{3} & +17x^{2} & -46x & +28 \\ \end{array} \] **Step 4:** Repeat the process with the new dividend \( 7x^{3} + 17x^{2} - 46x + 28 \). \[ \frac{7x^{3}}{x} = 7x^{2} \] Multiply and subtract: \[ (x - 7) \times 7x^{2} = 7x^{3} - 49x^{2} \] \[ \begin{array}{r|llll} & 0 & +7x^{3} & +17x^{2} & -46x & +28 \\ \hline & & 7x^{3} & -49x^{2} & & \\ \hline & & 0 & +66x^{2} & -46x & +28 \\ \end{array} \] **Step 5:** Continue with \( 66x^{2} - 46x + 28 \). \[ \frac{66x^{2}}{x} = 66x \] Multiply and subtract: \[ (x - 7) \times 66x = 66x^{2} - 462x \] \[ \begin{array}{r|llll} & & 0 & +66x^{2} & -46x & +28 \\ \hline & & & 66x^{2} & -462x & \\ \hline & & & 0 & +416x & +28 \\ \end{array} \] **Step 6:** Proceed with \( 416x + 28 \). \[ \frac{416x}{x} = 416 \] Multiply and subtract: \[ (x - 7) \times 416 = 416x - 2912 \] \[ \begin{array}{r|llll} & & & 0 & +416x & +28 \\ \hline & & & & 416x & -2912 \\ \hline & & & & 0 & +2940 \\ \end{array} \] **Step 7:** Since the degree of the remainder \( 2940 \) is less than the degree of the divisor \( x - 7 \), the division process stops here. --- ### Final Result The result of the division is: \[ \frac{x^{4} + 17x^{2} - 46x + 28}{x - 7} = x^{3} + 7x^{2} + 66x + 416 + \frac{2940}{x - 7} \] Or, more compactly: \[ x^{4} + 17x^{2} - 46x + 28 = (x - 7)(x^{3} + 7x^{2} + 66x + 416) + 2940 \] **Interpretation:** - **Quotient:** \( x^{3} + 7x^{2} + 66x + 416 \) - **Remainder:** \( 2940 \) --- **Note:** If the original polynomial was intended to include an \( x^{3} \) term (for example, \( x^{4} - 10x^{3} + 27x^{2} - 46x + 28 \)), please let me know so I can adjust the division accordingly.