Determine factor for $ f(x)=x^3+b^3$

$$
(fx)=x^3+b^3=(x+b)^3
$$

Substitute the value of $x=-b$ as follows:

$$
f(-b)=-b^3+b^3 f(-b)=0
$$

So the $(x+b)$ is a factor.

$$
text{color{Brown}$(x+b)$}
$$

Factor the following expression.

$$
color{#4257b2}text{(a)} x^3-64
$$

$$
(x^3-4^3)=x^3-64
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
x^3-4^3=(x-4)(x^2+4x+16)
$$

$$
color{#4257b2}text{(b)} 8x^3-27
$$

$$
[(2x)^3-3^3]=8x^3-27
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(2x)^3-3^3=(2x-3)(4x^2+6x+9)
$$

$$
color{#4257b2}text{(c)} 27x^3+8
$$

$$
[(3x)^3+2^3]=27x^3+8
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(3x)^3+2^3=(3x+2)(9x^2-6x+4)
$$

$$
color{#4257b2}text{(d)} x^3-125
$$

$$
[(x)^3-5^3]=x^3-125
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(x)^3-5^3=(x-5)(x^2+5x+25)
$$

$$
color{#4257b2}text{(e)} 64x^3-125
$$

$$
[(4x)^3-5^3]=64x^3-125
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(4x)^3-5^3=(4x-5)(16x^2+20x+25)
$$

$$
color{#4257b2}text{(f)} 1000x^3+729
$$

$$
[(10x)^3+9^3]=1000x^3+729
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(10x)^3+9^3=(10x+9)(100x^2-90x+81)
$$

$$
color{#4257b2}text{(g)} x^3+8
$$

$$
(x^3+2^3)=x^3+8
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
x^3+2^3=(x+2)(x^2-2x+4)
$$

$$
color{#4257b2}text{(h)} x^3+1
$$

$$
[(x)^3+1^3]=x^3+1
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(x)^3+1^3=(x+1)(x^2-x+1)
$$

$$
color{#4257b2}text{(i)} 216x^3-8
$$

$$
[(6x)^3-2^3]=x^3-8
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(6x)^3-2^3=(6x-2)(36x^2+12x+4)
$$

$$
text{color{Brown}(a) $x^3-4^3=(x-4)(x^2+4x+16)$ (b) $(2x-3)(4x^2+6x+9)$
\ \
(c) $(3x+2)(9x^2-6x+4)$ (d) $(x-5)(x^2+5x+25)$
\ \
(e) $(4x-5)(16x^2+20x+25)$ (f) $(10x+9)(100x^2-90x+81)$
\ \
(g) $(x+2)(x^2-2x+4)$ (h) $(x+1)(x^2-x+1)$
\ \
(i) $(6x-2)(36x^2+12x+4)$}
$$

Factor the following expression.

$$
color{#4257b2}text{(a)} 64x^3+27y^3
$$

$$
((4x)^3+(3y)^3)=64x^3+27y^3
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(4x)^3+(3y)^3=(4x+3y)(16x^2-12xy+9y^2)
$$

$$
color{#4257b2}text{(b)} -3x^4+24x
$$

Use common factor as follows:

$$
-3x(x^3-8)
$$

$$
-3x[x^3-2^3]=-3x(x^3-8)
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
-3x[x^3-2^3]=-3x (x-2)(x^2+2x+4)
$$

$$
color{#4257b2}text{(c)} (x+5)^3-(2x+1)^3
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(x+5)^3-(2x+1)^3=[(x+5)-(2x+1)][(x+5)^2+(x+5)(2x+1)+(2x+1)^2]
$$

$$
(x+5)^3-(2x+1)^3=[x+5-2x-1][x^2+10x+25+2x^2+x+10x+5+4x^2+4x+1]
$$

$$
(x+5)^3-(2x+1)^3=(-x+4)(7x^+25x+31)
$$

$$
color{#4257b2}text{(d)} x^6+64
$$

$$
[((x)^2)^3+4^3]=x^6+64
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
[(x)^2]^3+4^3=(x^2+4)(x^4-4x^2+1+16)
$$

$$
text{color{Brown}(a) $(4x+3y)(16x^2-12xy+9y^2)$
\ \
(b) $-3x (x-2)(x^2+2x+4)$
\ \
(c) $(-x+4)(7x^+25x+31)$
\ \
(d) $(x^2+4)(x^4-4x^2+1+16)$}
$$

Factor the following expression.

$$
color{#4257b2}text{(a)} x^3-343
$$

$$
(x^3-7^3)=x^3-343
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
x^3-7^3=(x-7)(x^2+7x+49)
$$

$$
color{#4257b2}text{(b)} 125x^3-512
$$

$$
((5x)^3-8^3)=125x^3-512
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(5x)^3-8^3=(5x-8)(25x^2+40x+64)
$$

$$
color{#4257b2}text{(c)} 512x^3+1
$$

$$
[(8x)^3+1^3]=512x^3+1
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(8x)^3+1^3=(8x+1)(64x^2-8x+1)
$$

$$
color{#4257b2}text{(d)} 216x^3-1
$$

$$
((6x)^3-1^3)=216x^3-1
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(6x)^3-1^3=(6x-1)(36x^2+6x+1)
$$

$$
color{#4257b2}text{(e)} 64x^3-1331
$$

$$
((4x)^3-11^3)=64x^3-1331
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
(4x)^3-11^3=(4x-11)(16x^2+44x+121)
$$

$$
color{#4257b2}text{(f)} 1331x^3+1728
$$

$$
((11x)^3+12^3)=1331x^3+1728
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(11x)^3+12^3=(11x+12)(121x^2-132x+144)
$$

$$
color{#4257b2}text{(g)} x^3+1000
$$

$$
(x^3+10^3)=x^3+1000
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
x^3+10^3=(x+10)(x^2-10x+100)
$$

$$
color{#4257b2}text{(h)} 343x^3+27
$$

$$
((7x)^3+3^3)=343x^3+27
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(7x)^3+3^3=(7x+3)(49x^2-21x+9)
$$

$$
color{#4257b2}text{(i)} 512-1331x^3
$$

$$
(8^3-(11x)^3)=512-1331x^3
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
8^3-(11x)^3=(8-11x)(64+88x+121x^2)
$$

$$
text{color{Brown}(a) $(x-7)(x^2+7x+49)$ (b) $(5x-8)(25x^2+40x+64)$
\ \
(c) $(8x+1)(64x^2-8x+1)$ (d) $(6x-1)(36x^2+6x+1)$
\ \
(e) $(4x-11)(16x^2+44x+121)$ (f) $(11x+12)(121x^2-132x+144)$
\ \
(g) $(x+10)(x^2-10x+100)$ (h) $(7x+3)(49x^2-21x+9)$
\ \
(i) $(8-11x)(64+88x+121x^2)$}
$$

Factor the following expression.

$$
color{#4257b2}text{(a)} dfrac{1}{27} x^3-dfrac{8}{125}
$$

$$
left[left(tfrac{1}{3}xright)^3-left(tfrac{2}{5}right)^3right]=tfrac{1}{27} x^3-tfrac{8}{125}
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
left[left(tfrac{1}{3}xright)^3-left(tfrac{2}{5}right)^3right]=left[left(tfrac{1}{3}xright)-left(tfrac{2}{5}right)right ] left[tfrac{1}{9}x^2+tfrac{2}{15}x+tfrac{4}{25}right]
$$

$$
color{#4257b2}text{(b)} -432x^5-128x^2
$$

Use common factor as follows:

$$
-2x^2(216x^3+64)=-2x^2[(6x)^3+6^3]
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
-2x^2[(6x)^3+6^3]=-2x^2(6x+6)(36x^2-36x+36)
$$

$$
color{#4257b2}text{(c)} (x-3)^3+(3x-2)^3
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(x-3)^3+(3x-2)^3=left[(x-3)+(3x-2)][(x-3)^2-(x-3)(3x-2)+(3x-2)^2right]
$$

$$
(x-3)^3+(3x-2)^3=[x-3+3x-2][x^2-6x+9-x^2+2x+9x-6+9x^2-12x+4]
$$

$$
(x-3)^3+(3x-2)^3=(4x-5)(9x^-7x+7)
$$

$$
color{#4257b2}text{(d)} dfrac{1}{512} x^9-512
$$

$$
left(tfrac{1}{8} x^3right)^3-8^3=tfrac{1}{512} x^9-512
$$

Use standard form $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$
left(tfrac{1}{8} x^3right)^3-8^3=left(tfrac{1}{8}x^3-8right)left(tfrac{1}{64} x^6+x^+364right)
$$

$$
text{color{Brown}(a) $left[left(tfrac{1}{3}xright)-left(tfrac{2}{5}right)right ] left(tfrac{1}{9}x^2+tfrac{2}{15}x+tfrac{4}{25}right)$
\ \ \
(b) $-2x^2(6x+6)(36x^2-36x+36)$
\ \ \
(c) $(4x-5)(9x^-7x+7)$
\ \ \
(d) $left(tfrac{1}{8}x^3-8right)left(tfrac{1}{64} x^6+x^+364right)$}
$$

Do you agree with Jarred or not.

$$
dfrac{(a+b)(a^2-ab+b^2)+(a-b)(a^2+ab+b^2)}{2a^3}=1
$$

Use distributive property as follows:

$$
dfrac{a^3-a^2b+ab^2+ba^2-ab^2+b^3+a^3+a^2b+ab^2-ba^2-ab^2-b^3}{2a^3}=1
$$

$$
dfrac{2a^3}{2a^3}=1
$$

The Jarred is correct.

$$
text{color{Brown}$$dfrac{2a^3}{2a^3}=1$$}
$$

Use factorization for the sum of cubes to verify that these sum are equal.

$$
color{#4257b2}1^3+12^3 9^3+10^3
$$

$$
1^3+12^3
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
1^3+12^3=(1+12)(1-12+144)
$$

$$
1^3+12^3=13(133)=1729
$$

$$
9^3+10^3
$$

Use standard form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
9^3+10^3=(9+10)(81-90+100)
$$

$$
9^3+10^3=19(91)=1729
$$

The sum of two cubes is equal.

$$
text{color{Brown}$$1^3+12^3=1729$$

$$9^3+10^3=1729$$}
$$

Prove that :

$$
color{#4257b2}(x^2+y^2)(x^4-x^2y^2+y^4)(x^{12}-x^6y^6+y^{12})+2x^9y^9=(x^9+y^9)^2
$$

According form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(x^2+y^2)(x^4-x^2y^2+y^4)=(x^2)^3+(y^2)^3
$$

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

$$
(x^6+y^6)(x^{12}-x^6y^6+y^{12})
$$

According form $a^3+b^3=(a+b)(a^2-ab+b^2)$

$$
(x^6+y^6)(x^{12}-x^6y^6+y^{12})=(x^6)^3+(y^6)^3
$$

$$
(x^6)^3+(y^6)^3=x^{18}+y^{18}
$$

$$
x^{18}+y^{18}+2x^9y^9=x^{18}+2x^9y^9+y^{18}=(x^9+y^9)^2
$$

The left side is equal the right side.

$$
text{color{Brown}The left side is equal the right side}
$$

When you need to factor the cubic function degree, you should follow the standard equation as follows:

$$
a^3+b^3=(a+b)(a^2-ab+b^2)
$$

$$
a^3-b^3=(a-b)(a^2+ab+b^2)
$$

$$
text{color{Brown}$$a^3+b^3=(a+b)(a^2-ab+b^2)$$

$$a^3-b^3=(a-b)(a^2+ab+b^2)$$}
$$

The number $1729$ is called taxicab number, so answer for the following questions>

(a) Why is called a taxicab number?

is defined as the smallest number that can be expressed as a sum of two positive cube numbers in n distinct ways.

The most famous taxicab number is $1729 = text{Ta} (2) = 1^3 + 12^3 = 9^3 + 10^3$

(b) Are there other taxicab number?

There are other six taxicab number such as:

$$
87539319 6963472309248
$$

$$
text{color{Brown}(a) Because expressed as a sum of two positive cubes \
numbers in (n) distinct ways.
\ \
(b) Are there other taxicab number?
\
$$87539319 6963472309248$$}
$$