Questions: Write the function in the form f(x)=(x-k) q(x)+r(x) for the given value of k. f(x)=-3 x^3+8 x^2+13 x-4, k=2+sqrt3 f(x)= Demonstrate that f(k)=r. f(2+sqrt3)=

Transcript text: Write the function in the form $f(x)=(x-k) q(x)+r(x)$ for the given value of $k$. \[ \begin{array}{l} f(x)=-3 x^{3}+8 x^{2}+13 x-4, k=2+\sqrt{3} \\ f(x)=\square \end{array} \] $\square$ Demonstrate that $f(k)=r$. \[ f(2+\sqrt{3})= \] $\square$

Solution

Solution Steps

Step 1: Understand the Problem

We need to express the polynomial function $ f(x) = -3x^3 + 8x^2 + 13x - 4 $ in the form $ f(x) = (x - k) q(x) + r(x) $ for $ k = 2 + \sqrt{3} $. Then, we need to demonstrate that $ f(k) = r $.

Step 2: Polynomial Division

To express $ f(x) $ in the desired form, we perform polynomial division of $ f(x) $ by $ x - (2 + \sqrt{3}) $.

Divide the leading term: \[ \frac{-3x^3}{x} = -3x^2 \]
Multiply and subtract: \[ (-3x^2)(x - (2 + \sqrt{3})) = -3x^3 + 3(2 + \sqrt{3})x^2 \] \[ -3x^3 + 8x^2 + 13x - 4 - (-3x^3 + 6x^2 + 3\sqrt{3}x^2) = 2x^2 - 3\sqrt{3}x^2 + 13x - 4 \]
Combine like terms: \[ 2x^2 - 3\sqrt{3}x^2 = (2 - 3\sqrt{3})x^2 \]
Repeat the process: \[ \frac{(2 - 3\sqrt{3})x^2}{x} = (2 - 3\sqrt{3})x \] \[ ((2 - 3\sqrt{3})x)(x - (2 + \sqrt{3})) = (2 - 3\sqrt{3})x^2 - (2 - 3\sqrt{3})(2 + \sqrt{3})x \] \[ (2 - 3\sqrt{3})x^2 - (4 + 2\sqrt{3} - 6\sqrt{3} - 9)x = (2 - 3\sqrt{3})x^2 - (4 - 4\sqrt{3} - 9)x \] \[ 2x^2 - 3\sqrt{3}x^2 + 13x - 4 - (2x^2 - 3\sqrt{3}x^2 - 4x + 4\sqrt{3}x + 9x) = 13x - 4 - (-4x + 4\sqrt{3}x + 9x) \] \[ 13x - 4 - (-4x + 4\sqrt{3}x + 9x) = 13x - 4 + 4x - 4\sqrt{3}x - 9x = (13 + 4 - 9)x - 4 - 4\sqrt{3}x = 8x - 4 - 4\sqrt{3}x \]
Combine like terms: \[ 8x - 4 - 4\sqrt{3}x = (8 - 4\sqrt{3})x - 4 \]
Repeat the process: \[ \frac{(8 - 4\sqrt{3})x}{x} = 8 - 4\sqrt{3} \] \[ (8 - 4\sqrt{3})(x - (2 + \sqrt{3})) = (8 - 4\sqrt{3})x - (8 - 4\sqrt{3})(2 + \sqrt{3}) \] \[ (8 - 4\sqrt{3})x - (16 + 8\sqrt{3} - 8\sqrt{3} - 12) = (8 - 4\sqrt{3})x - 4 \] \[ 8x - 4 - 4\sqrt{3}x - 4 = 0 \]

Step 3: Verify $ f(k) = r $

Substitute $ k = 2 + \sqrt{3} $ into $ f(x) $: \[ f(2 + \sqrt{3}) = -3(2 + \sqrt{3})^3 + 8(2 + \sqrt{3})^2 + 13(2 + \sqrt{3}) - 4 \] Calculate each term: \[ (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \] \[ (2 + \sqrt{3})^3 = (2 + \sqrt{3})(7 + 4\sqrt{3}) = 14 + 8\sqrt{3} + 7\sqrt{3} + 12 = 26 + 15\sqrt{3} \] \[ -3(2 + \sqrt{3})^3 = -3(26 + 15\sqrt{3}) = -78 - 45\sqrt{3} \] \[ 8(2 + \sqrt{3})^2 = 8(7 + 4\sqrt{3}) = 56 + 32\sqrt{3} \] \[ 13(2 + \sqrt{3}) = 26 + 13\sqrt{3} \] Combine all terms: \[ f(2 + \sqrt{3}) = -78 - 45\sqrt{3} + 56 + 32\sqrt{3} + 26 + 13\sqrt{3} - 4 \] \[ = -78 + 56 + 26 - 4 + (-45\sqrt{3} + 32\sqrt{3} + 13\sqrt{3}) \] \[ = 0 \]