# Math Help - sum and differences? 9th grade?

1. ## sum and differences? 9th grade?

Use these patterns to factor polynomials that are sums and differences of cubes.
a^3 + b^3 = (a+b)(a^2-ab+b^2)
a^3-b^3=(a-b)(a^2+ab+b^2)

a)x+8
b)8x^3+27
c)64-y^3
d)2u^3 + 16V^3
e)d^5 + f^3
f) w^6 -1

2. if the sum of two numbers is 6 and the sum of their cubes is 18, what's the sum of their squares?

3. use synthetic division to find (x^2 + 4) / (s - 2i)

Please show work! Thank you so so much!!!!

2. Originally Posted by tecktonikk
Use these patterns to factor polynomials that are sums and differences of cubes.
a^3 + b^3 = (a+b)(a^2-ab+b^2)
a^3-b^3=(a-b)(a^2+ab+b^2)

a)x+8 $\textcolor{red}{(\sqrt[3]{x})^2 + 2^3}$

b)8x^3+27 $\textcolor{red}{(2x)^3 + 3^3}$

c)64-y^3 $\textcolor{red}{4^3 - y^3}$

d)2u^3 + 16V^3 $\textcolor{red}{2[u^3 + (2v)^3]}$

e)d^5 + f^3 $\textcolor{red}{(d^{\frac{5}{3}})^3 + f^3}$

f) w^6 -1 $\textcolor{red}{(w^2)^3 - 1^3}$

2. if the sum of two numbers is 6 and the sum of their cubes is 18, what's the sum of their squares?

$\textcolor{red}{x+y = 6}$

$\textcolor{red}{x^3+y^3 = 18}$

also, note that $\textcolor{red}{(x+y)^2 = 36}$

3. use synthetic division to find (x^2 + 4) / (s - 2i)
Code:
2i] 1 ..... 0 ...... 4

----------------------
all set up ... you do the work .

3. Hello, tecktonikk!

2. If the sum of two numbers is 6 and the sum of their cubes is 18,
what is the sum of their squares?
Let the two numbers be $x$ and $y$.

We have: . $\begin{array}{cccc}x+y &=& 6 & {\color{blue}[1]} \\ x^3+y^3 &=& 18 & {\color{blue}[2]} \end{array}$

Cube equation [1]: . $(x+y)^3 \:=\:6^3 \quad\Rightarrow\quad x^3 + 3x^2y + 3xy^2 + y^3 \:=\:216$

$\text{We have: }\;\underbrace{x^3 + y^3}_{18} + \; 3xy\underbrace{(x+y)}_6 \;=\;216 \quad\Rightarrow\quad 18 \;+\; 18xy \:=\:216 \quad\Rightarrow\quad xy \:=\:11$

Square equation [1]: . $(x+y)^2 \:=\:6^2 \quad\Rightarrow\quad x^2 + 2xy + y^2 \:=\:36$

$\text{We have: }\;x^2 + y^2 + 2\underbrace{(xy)}_{11} \:=\:36 \quad\Rightarrow\quad x^2 + y^2 + 22 \:=\:36 \quad\Rightarrow\quad\boxed{x^2+y^2 \:=\:14}$