Find the equation of the line of the below a) gradient 2 passes through (0,8) b) gradient 4 passes through (0,3) 5. Divide a) (2x³4x² + 6x - 3) ÷ (x+4) b) (6x³ - 4x² + 3x + 2) ÷ (2x + 2)

Here are the solutions to the problems. 4. Find the equation of the line The general equation of a line is y = mx + c, where m is the gradient (slope) and c is the y-intercept. a) gradient 2 passes through (0, 8) Step 1: Identify the gradient and the y-intercept. The gradient is given as m = 2. The line passes through the point (0, 8). Since the x-coordinate is 0, this point is the y-intercept, so c = 8. Step 2: Substitute the values into the equation y = mx + c. y = 2x + 8 The equation of the line is y = 2x + 8. b) gradient 4 passes through (0, 3) Step 1: Identify the gradient and the y-intercept. The gradient is given as m = 4. The line passes through the point (0, 3). Since the x-coordinate is 0, this point is the y-intercept, so c = 3. Step 2: Substitute the values into the equation y = mx + c. y = 4x + 3 The equation of the line is y = 4x + 3. 5. Divide the polynomials a) (2x^3 4x^2 + 6x - 3) ÷ (x + 4) Assumption: The term 2x^3 4x^2 is interpreted as 2x^3 + 4x^2. If it was intended as a product, the problem would be significantly more complex and usually written with a multiplication symbol. We will perform polynomial long division for (2x^3 + 4x^2 + 6x - 3) ÷ (x + 4). Step 1: Divide 2x^3 by x. c|cc cc 2r2x^2 & & & \\ 2-5 x+4 & 2x^3 & +4x^2 & +6x & -3 \\ 2r-(2x^3 & +8x^2) & & \\ 2-3 2r0 & -4x^2 & +6x & \\ Step 2: Divide -4x^2 by x. c|cc cc 2r2x^2 & -4x & & \\ 2-5 x+4 & 2x^3 & +4x^2 & +6x & -3 \\ 2r-(2x^3 & +8x^2) & & \\ 2-3 2r0 & -4x^2 & +6x & \\ 2r & -(-4x^2 & -16x) & \\ 3-4 2r & 0 & 22x & -3 \\ Step 3: Divide 22x by x. c|cc cc 2r2x^2 & -4x & +22 & \\ 2-5 x+4 & 2x^3 & +4x^2 & +6x & -3 \\ 2r-(2x^3 & +8x^2) & & \\ 2-3 2r0 & -4x^2 & +6x & \\ 2r & -(-4x^2 & -16x) & \\ 3-4 2r & 0 & 22x & -3 \\ 2r & & -(22x & +88) \\ 4-5 2r & & 0 & -91 \\ The quotient is 2x^2 - 4x + 22 and the remainder is -91. Therefore, (2x^3 + 4x^2 + 6x - 3) ÷ (x + 4) = 2x^2 - 4x + 22 - (91)/(x+4). The result of the division is 2x^2 - 4x + 22 - (91)/(x+4). b) (6x^3 - 4x^2 + 3x + 2) ÷ (2x + 2) We will perform polynomial long division for (6x^3 - 4x^2 + 3x + 2) ÷ (2x + 2). Step 1: Divide 6x^3 by 2x. c|cc cc 2r3x^2 & & & \\ 2-5 2x+2 & 6x^3 & -4x^2 & +3x & +2 \\ 2r-(6x^3 & +6x^2) & & \\ 2-3 2r0 & -10x^2 & +3x & \\ Step 2: Divide -10x^2 by 2x. c|cc cc 2r3x^2 & -5x & & \\ 2-5 2x+2 & 6x^3 & -4x^2 & +3x & +2 \\ 2r-(6x^3 & +6x^2) & & \\ 2-3 2r0 & -10x^2 & +3x & \\ 2r & -(-10x^2 & -10x) & \\ 3-4 2r & 0 & 13x & +2 \\ Step 3: Divide 13x by 2x. c|cc cc 2r3x^2 & -5x & +(13)/(2) & \\ 2-5 2x+2 & 6x^3 & -4x^2 & +3x & +2 \\ 2r-(6x^3 & +6x^2) & & \\ 2-3 2r0 & -10x^2 & +3x & \\ 2r & -(-10x^2 & -10x) & \\ 3-4 2r & 0 & 13x & +2 \\ 2r & & -(13x & +13) \\ 4-5 2r & & 0 & -11 \\ The quotient is 3x^2 - 5x + (13)/(2) and the remainder is -11. Therefore, (6x^3 - 4x^2 + 3x + 2) ÷ (2x + 2) = 3x^2 - 5x + (13)/(2) - (11)/(2x+2). The result of the division is 3x^2 - 5x + (13)/(2) - (11)/(2x+2).