1. Problem: Find the value of $f(3)$ for the polynomial function $f(x) = x^4 - 3x^2 + 4x + 2$ using substitution and Horner's method. 2. Formula and rules: Horner's method is a synthetic division technique to evaluate polynomials efficiently. For $f(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0$, we write coefficients and perform synthetic division by $x - c$ where $c=3$. 3. Coefficients: $[1, 0, -3, 4, 2]$ (note $x^3$ term coefficient is 0). 4. Horner's method steps: - Bring down 1. - Multiply 1 by 3: 3, add to 0: 3. - Multiply 3 by 3: 9, add to -3: 6. - Multiply 6 by 3: 18, add to 4: 22. - Multiply 22 by 3: 66, add to 2: 68. 5. Result: $f(3) = 68$. --- 1. Problem: Find quotient and remainder of dividing $x^4 - 3x^2 + 5x - 3$ by $x - 2$. 2. Use synthetic division with $c=2$ and coefficients $[1, 0, -3, 5, -3]$. 3. Steps: - Bring down 1. - Multiply 1 by 2: 2, add to 0: 2. - Multiply 2 by 2: 4, add to -3: 1. - Multiply 1 by 2: 2, add to 5: 7. - Multiply 7 by 2: 14, add to -3: 11. 4. Quotient coefficients: $1, 2, 1, 7$ corresponding to $x^3 + 2x^2 + x + 7$. 5. Remainder: 11. --- 1. Problem: Sketch the graph of the rational function $y = \frac{x + 4}{x - 2}$. 2. Important features: - Vertical asymptote at $x=2$ (denominator zero). - Horizontal asymptote: degrees numerator and denominator are equal, so $y = \frac{1}{1} = 1$. 3. Behavior: - As $x \to 2^-$, $y \to -\infty$ or $+\infty$ depending on numerator. - As $x \to \pm \infty$, $y \to 1$. 4. The graph is a hyperbola with two branches separated by the vertical asymptote. Final answers: - $f(3) = 68$ - Quotient: $x^3 + 2x^2 + x + 7$, Remainder: 11 - Graph: $y = \frac{x + 4}{x - 2}$ with vertical asymptote at $x=2$ and horizontal asymptote at $y=1$