1. **State the problem:** Find the quotient and remainder when dividing the polynomial $x^4 - 16x^3 + 4x^2 + 10x - 11$ by a divisor (not specified here, so we focus on understanding the polynomial). 2. **Degree of quotient for each division:** - a) Dividend degree = 4, Divisor degree = 2, Quotient degree = $4 - 2 = 2$ - b) Dividend degree = 3, Divisor degree = 1, Quotient degree = $3 - 1 = 2$ - c) Dividend degree = 4, Divisor degree = 3, Quotient degree = $4 - 3 = 1$ - d) Dividend degree = 2, Divisor degree = 4, Quotient degree = $2 - 4 = -2$ (not possible, so no quotient) 3. **Complete the divisions if possible:** **a) Divide** $x^4 - 15x^3 + 2x^2 + 12x - 10$ by $x^2 - 4$: 1. Divide leading terms: $\frac{x^4}{x^2} = x^2$ 2. Multiply divisor by $x^2$: $x^2(x^2 - 4) = x^4 - 4x^2$ 3. Subtract: $(x^4 - 15x^3 + 2x^2 + 12x - 10) - (x^4 - 4x^2) = -15x^3 + 6x^2 + 12x - 10$ 4. Divide leading terms: $\frac{-15x^3}{x^2} = -15x$ 5. Multiply divisor by $-15x$: $-15x(x^2 - 4) = -15x^3 + 60x$ 6. Subtract: $(-15x^3 + 6x^2 + 12x - 10) - (-15x^3 + 60x) = 6x^2 - 48x - 10$ 7. Divide leading terms: $\frac{6x^2}{x^2} = 6$ 8. Multiply divisor by $6$: $6(x^2 - 4) = 6x^2 - 24$ 9. Subtract: $(6x^2 - 48x - 10) - (6x^2 - 24) = -48x + 14$ **Quotient:** $x^2 - 15x + 6$ **Remainder:** $-48x + 14$ **b) Divide** $5x^3 - 4x^2 + 3x - 4$ by $x + 3$: 1. Divide leading terms: $\frac{5x^3}{x} = 5x^2$ 2. Multiply divisor by $5x^2$: $5x^2(x + 3) = 5x^3 + 15x^2$ 3. Subtract: $(5x^3 - 4x^2 + 3x - 4) - (5x^3 + 15x^2) = -19x^2 + 3x - 4$ 4. Divide leading terms: $\frac{-19x^2}{x} = -19x$ 5. Multiply divisor by $-19x$: $-19x(x + 3) = -19x^2 - 57x$ 6. Subtract: $(-19x^2 + 3x - 4) - (-19x^2 - 57x) = 60x - 4$ 7. Divide leading terms: $\frac{60x}{x} = 60$ 8. Multiply divisor by $60$: $60(x + 3) = 60x + 180$ 9. Subtract: $(60x - 4) - (60x + 180) = -184$ **Quotient:** $5x^2 - 19x + 60$ **Remainder:** $-184$ **c) Divide** $x^4 - 7x^3 + 2x^2 + 9x$ by $x^3 - x^2 + 2x + 1$: 1. Divide leading terms: $\frac{x^4}{x^3} = x$ 2. Multiply divisor by $x$: $x(x^3 - x^2 + 2x + 1) = x^4 - x^3 + 2x^2 + x$ 3. Subtract: $(x^4 - 7x^3 + 2x^2 + 9x) - (x^4 - x^3 + 2x^2 + x) = -6x^3 + 8x$ Degree of remainder is 3, same as divisor degree, so continue: 4. Divide leading terms: $\frac{-6x^3}{x^3} = -6$ 5. Multiply divisor by $-6$: $-6(x^3 - x^2 + 2x + 1) = -6x^3 + 6x^2 - 12x - 6$ 6. Subtract: $(-6x^3 + 8x) - (-6x^3 + 6x^2 - 12x - 6) = -6x^2 + 20x + 6$ Degree of remainder is 2, less than divisor degree 3, stop. **Quotient:** $x - 6$ **Remainder:** $-6x^2 + 20x + 6$ **d) Divide** $2x^2 + 5x - 4$ by $x^4 + 3x^3 - 5x^2 + 4x - 2$: Degree of dividend (2) is less than divisor (4), so quotient is 0 and remainder is dividend. 4. **Complete the table:** | Dividend | Divisor | Quotient | Remainder | |--------------------------|---------|---------------------|-----------| | $2x^3 - 5x^2 + 8x + 4$ | $x + 3$ | $2x^2 - 11x + 41$ | $-119$ | | $3x^3 - 5x + 8$ | $2x + 4$ | $\frac{3}{2}x^2 - 3x + \frac{1}{2}$ | $-3$ | | $6x^4 + 2x^3 + 3x^2 - 11x - 9$ | (not given) | $2x^3 + x - 4$ | $-5$ | | $3x^3 + x^2 - 6x + 16$ | $x + 2$ | $3x^2 - 5x + 4$ | $8$ | **Note:** For the first row remainder, synthetic division or polynomial division shows remainder $-119$. --- **Final answers:** - Degree of quotients: a) 2, b) 2, c) 1, d) no quotient - Completed divisions for a), b), c), d) as above - Completed table with quotient and remainder values