Question

additional exercises 6.4
form 1
division of polynomials
divide.

1. $\frac{4x^{10}+10x^{6}}{2x^{2}}$
2. $\frac{12x^{2}-18x^{4}}{3x^{2}}$
3. $\frac{49x^{6}-35x^{5}+49x^{4}}{7x^{3}}$
4. $\frac{7x^{5}-14x^{3}+21x^{2}}{-7x^{2}}$
5. $(14x^{7}-8x^{5}+8x^{3})\div(2x^{3})$
6. $(42x^{7}-48x^{6}+36x^{4}+30x^{2})\div(6x^{4})$
7. $(x^{2}-25)\div(x+5)$
8. $(x^{2}+5x+6)\div(x+3)$

Explanation:

Problem 1

Step1: Split the fraction

$\frac{4x^{10}}{2x^2} + \frac{10x^6}{2x^2}$

Step2: Simplify each term

$2x^{10-2} + 5x^{6-2} = 2x^8 + 5x^4$

Problem 2

Step1: Split the fraction

$\frac{12x^2}{3x^2} - \frac{18x^4}{3x^2}$

Step2: Simplify each term

$4 - 6x^{4-2} = 4 - 6x^2$

Problem 3

Step1: Split the fraction

$\frac{49x^6}{7x^3} - \frac{35x^3}{7x^3} + \frac{49x^4}{7x^3}$

Step2: Simplify each term

$7x^{6-3} - 5 + 7x^{4-3} = 7x^3 - 5 + 7x$

Problem 4

Step1: Split the fraction

$\frac{7x^5}{-7x^2} - \frac{14x^3}{-7x^2} + \frac{21x^2}{-7x^2}$

Step2: Simplify each term

$-x^{5-2} + 2x^{3-2} - 3 = -x^3 + 2x - 3$

Problem 5

Step1: Split the division

$\frac{14x^7}{2x^3} - \frac{8x^5}{2x^3} + \frac{8x^3}{2x^3}$

Step2: Simplify each term

$7x^{7-3} - 4x^{5-3} + 4 = 7x^4 - 4x^2 + 4$

Problem 6

Step1: Split the division

$\frac{42x^7}{6x^4} - \frac{48x^6}{6x^4} + \frac{36x^4}{6x^4} + \frac{30x^2}{6x^4}$

Step2: Simplify each term

$7x^{7-4} - 8x^{6-4} + 6 + 5x^{2-4} = 7x^3 - 8x^2 + 6 + \frac{5}{x^2}$

Problem 8

Step1: Factor the numerator

$x^2 -25 = (x+5)(x-5)$

Step2: Cancel common factors

$\frac{(x+5)(x-5)}{x+5} = x-5$

Problem 10

Step1: Factor the numerator

$x^2+5x+6 = (x+3)(x+2)$

Step2: Cancel common factors

$\frac{(x+3)(x+2)}{x+3} = x+2$

Answer:

1. $2x^8 + 5x^4$
2. $4 - 6x^2$
3. $7x^3 + 7x - 5$
4. $-x^3 + 2x - 3$
5. $7x^4 - 4x^2 + 4$
6. $7x^3 - 8x^2 + 6 + \frac{5}{x^2}$
7. $x - 5$
8. $x + 2$

(Note: Problems 7 and 9 are not visible in the provided image, so no solution is given for them.)