3. Using synthetic division find the quotient and the remainder when i) \( x^{3}-2 x^{2}+9 \) is divided by \( (x+2) \) \( \begin{array}{llll}\text { iii) } 4 x^{7}+3 \text { is divided by } x-3 & \text { i) } x^{4}-2 x^{3}-3 x^{2}-4 x-8 \text { is divided by } x-2\end{array} \) 4. Factorize each of the following polynomials: \( \begin{array}{llll}\text { i) } x^{3}-2 x^{2}-5 x+6 & \text { ii) } x^{4}-1 & \text { iii) } x^{4}-2 x^{3}+x-2\end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(x^{4}-1=0\) - step1: Move the constant to the right side: \(x^{4}=0+1\) - step2: Remove 0: \(x^{4}=1\) - step3: Simplify the expression: \(x=\pm \sqrt[4]{1}\) - step4: Simplify: \(x=\pm 1\) - step5: Separate into possible cases: \(\begin{align}&x=1\\&x=-1\end{align}\) - step6: Rewrite: \(x_{1}=-1,x_{2}=1\) Solve the equation \( x^{3}-2 x^{2}-5 x+6=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(x^{3}-2x^{2}-5x+6=0\) - step1: Factor the expression: \(\left(x-3\right)\left(x-1\right)\left(x+2\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-3=0\\&x-1=0\\&x+2=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=3\\&x=1\\&x=-2\end{align}\) - step4: Rewrite: \(x_{1}=-2,x_{2}=1,x_{3}=3\) Solve the equation \( 4 x^{7}+3=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4x^{7}+3=0\) - step1: Move the constant to the right side: \(4x^{7}=0-3\) - step2: Remove 0: \(4x^{7}=-3\) - step3: Divide both sides: \(\frac{4x^{7}}{4}=\frac{-3}{4}\) - step4: Divide the numbers: \(x^{7}=-\frac{3}{4}\) - step5: Simplify the equation: \(\sqrt[7]{x^{7}}=\sqrt[7]{-\frac{3}{4}}\) - step6: Calculate: \(x=\sqrt[7]{-\frac{3}{4}}\) - step7: Simplify the root: \(x=-\frac{\sqrt[7]{96}}{2}\) Solve the equation \( x^{4}-2 x^{3}+x-2=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(x^{4}-2x^{3}+x-2=0\) - step1: Factor the expression: \(\left(x-2\right)\left(x+1\right)\left(x^{2}-x+1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-2=0\\&x+1=0\\&x^{2}-x+1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=2\\&x=-1\\&x \notin \mathbb{R}\end{align}\) - step4: Find the union: \(\begin{align}&x=2\\&x=-1\end{align}\) - step5: Rewrite: \(x_{1}=-1,x_{2}=2\) Solve the equation \( x^{3}-2 x^{2}+9=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve for \(x\) in the complex numbers system: \(x^{3}-2x^{2}+9=0\) - step1: Calculate: \(\begin{align}&x\approx -1.584544\\&x\approx 1.792272+1.570869i\\&x\approx 1.792272-1.570869i\end{align}\) - step2: Rewrite: \(x_{1}\approx 1.792272-1.570869i,x_{2}\approx 1.792272+1.570869i,x_{3}\approx -1.584544\) - step3: Remove the complex number(s): \(x\approx -1.584544\) Solve the equation \( x^{4}-2 x^{3}-3 x^{2}-4 x-8=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve for \(x\) in the complex numbers system: \(x^{4}-2x^{3}-3x^{2}-4x-8=0\) - step1: Calculate: \(\begin{align}&x\approx 3.419492\\&x\approx -0.040039-1.321014i\\&x\approx -0.040039+1.321014i\\&x\approx -1.339414\end{align}\) - step2: Rewrite: \(x_{1}\approx -1.339414,x_{2}\approx -0.040039+1.321014i,x_{3}\approx -0.040039-1.321014i,x_{4}\approx 3.419492\) - step3: Remove the complex number(s): \(x_{1}\approx -1.339414,x_{2}\approx 3.419492\) Let's solve the problems step by step. ### 3. Synthetic Division #### i) \( x^{3}-2x^{2}+9 \) divided by \( (x+2) \) To perform synthetic division, we will use the root of \( x + 2 = 0 \), which is \( x = -2 \). 1. Write the coefficients of the polynomial: \( 1, -2, 0, 9 \) (note that the coefficient of \( x \) is 0). 2. Set up synthetic division: ``` -2 | 1 -2 0 9 | -2 8 -16 --------------------- 1 -4 8 -7 ``` The bottom row gives us the coefficients of the quotient and the remainder: - Quotient: \( x^{2} - 4x + 8 \) - Remainder: \( -7 \) #### iii) \( 4x^{7}+3 \) divided by \( x-3 \) Using \( x = 3 \): 1. Write the coefficients: \( 4, 0, 0, 0, 0, 0, 3 \). 2. Set up synthetic division: ``` 3 | 4 0 0 0 0 0 3 | 12 36 108 324 972 --------------------------- 4 12 36 108 324 975 ``` The bottom row gives us: - Quotient: \( 4x^{6} + 12x^{5} + 36x^{4} + 108x^{3} + 324x^{2} + 975 \) - Remainder: \( 975 \) #### ii) \( x^{4}-2x^{3}-3x^{2}-4x-8 \) divided by \( x-2 \) Using \( x = 2 \): 1. Write the coefficients: \( 1, -2, -3, -4, -8 \). 2. Set up synthetic division: ``` 2 | 1 -2 -3 -4 -8 | 2 0 -6 -8 ------------------------- 1 0 -3 -10 -16 ``` The bottom row gives us: - Quotient: \( x^{3} - 3x - 10 \) - Remainder: \( -16 \) ### 4. Factorization #### i) \( x^{3}-2x^{2}-5x+6 \) Using the roots found earlier: - Roots: \( x_{1} = -2, x_{2} = 1, x_{3} = 3 \) Thus, the factorization is: \[ (x + 2)(x - 1)(x - 3) \] #### ii) \( x^{4}-1 \) This is a difference of squares: \[ x^{4}-1 = (x^{2}-1)(x^{2}+1) = (x-1)(x+1)(x^{2}+1) \] #### iii) \( x^{4}-2x^{3}+x-2 \) Using the roots found earlier: - Roots: \( x_{1} = -1, x_{2} = 2 \) Thus, the factorization is: \[ (x + 1)(x - 2)(x^{2} + ax + b) \] To find \( a \) and \( b \), we can use polynomial long division or synthetic division to find the quadratic factor. ### Summary of Results 1. **Synthetic Division Results:** - \( x^{3}-2x^{2}+9 \) by \( (x+2) \): Quotient \( x^{2}-4x+8 \), Remainder \( -7 \) - \( 4x^{7}+3 \) by \( (x-3) \): Quotient \( 4x^{6}+12x^{5}+36x^{4}+108x^{3}+324x^{2}+975 \), Remainder \( 975 \) - \( x^{4}-2x^{3}-3x^{2}-4x-8 \) by \( (x-2) \): Quotient \( x^{3}-3x-10 \), Remainder \( -16 \) 2. **Factorizations:** - \( x^{3}-2x^{2}-5x+6 = (x + 2)(x - 1)(x - 3) \) - \( x^{4}-1 = (x - 1)(x + 1)(x^{2} + 1) \) - \( x^{4}-2x^{3}+x-2 = (x + 1)(x - 2)(x^{2} + ax + b) \) (further factorization needed for the quadratic).