Question

1.Divide the polynomials using the division method of your choice (long or synthetic)

3x³ - 4x² - + 2x -1 / x - 3

2.Perform the indicated operation for the following:

a. (3x² - 8x + 4) - (6x² + 7x -1)

b. (4x⁷ - 3x⁵ - + 2x + 4) + (12x⁵ - 3x + 2x² + 3)

c. (3x + 4)(5x² - 2x + 3)

d. (3 + 4i) + (1-2i)

e. (2-i) (5-6i)

f. (3-7i) ( 1-2i)

3.Solve x² + 81= 0

4.Solve ³ square root of 7x = 3

Answer 1

Calculations

1)

• For this question refer to the figure attached in which both synthetic & long divisions are performed.

• The final answer is gained the same in both ways as

$\qquad\qquad \begin{aligned} \small \frac{3x^3-4x^2+2x-1}{(x-3)}&=\small (3x^2 +5x+17)+\frac{50}{(x-3)} \end{aligned}$

2)

a.$\qquad\qquad \begin{aligned} \small &=\small 3x^2-8x+4-(6x^2+7x-1)\\ &=\small 3x^2-8x+4-6x^2-7x+1\\ &=\small -3x^2-15x+5 \end{aligned}$

b.$\qquad\qquad \begin{aligned} &=\small 4x^7-3x^5+2x+4+(12x^5-3x+2x^2+3)\\ &=\small 4x^7-3x^5+2x+4+12x^5-3x+2x^2+3\\ &=\small 4x^7+9x^5+2x^2-x+7 \end{aligned}$

c.$\qquad\qquad \begin{aligned} \small &=\small 3x(5x^2-2x+3)+4(5x^2-2x+3)\\ &=\small 15x^3-6x^2+9x+20x^2-8x+12\\ &=\small 15x^3+14x^2+x+12 \end{aligned}$

d.$\qquad\qquad \begin{aligned} &=\small 3+4i+(1-2i)\\ &=\small 3+4i+1-2i\\ &=\small 4+2i \end{aligned}$

e.$\qquad\qquad \begin{aligned} &=\small (2-i)(5-6i)\\ &=\small 10-12i-5i+6i^2\\ &=\small 10-17i-6\\ &=\small 4-17i \end{aligned}$

f.$\qquad\qquad \begin{aligned} &=\small (3-7i)(1-2i)\\ &=\small 3-6i-7i+14(-1)\\ &=\small -11-13i \end{aligned}$

3)

$\qquad\qquad \begin{aligned} \small x^2-(-81)&=\small 0\\ \small x^2-81i^2&=\small 0\\ \small (x+9i)(x-9i)&=\small 0\\ \small x&=\small \pm9i \end{aligned}$

4)

$\qquad\qquad \begin{aligned} \small \sqrt[3]{7x}&=\small 3\\ \small (\sqrt[3]{7x})^3&=\small 3^3\\ \small 7x&=\small 27\\ \small x&=\small \frac{27}{7} \end{aligned}$