Question

check your compound word below:

1. $27x^3 - 1$ keyhole
2. $x^3 + 27$ paperclip
3. $x^3 - 64$ household
4. $125x^3 + 27$
5. $x^3 + 125$
6. $x^3 - 27$

check your compound word below:

1. $8x^3 + 125$
2. $64x^3 + 1$
3. $27x^3 - 8$
4. $8x^3 + 1$
5. $x^3 + 8y^3$
6. $64x^3 + y^3$

then there is a chart with checkboxes and expressions like $(x^2 + 3x + 9)$, $(16x^2 - 4x + 1)$, etc., and words like work, board, clip, paper, hand, news, letter, key, moon, dew, shake, guess, catcher, hole, hold, works, out, light, fire, black.

Explanation:

We use the sum and difference of cubes formulas:
Sum of cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
Difference of cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Step1: Factor $27x^3 -1$

$27x^3=(3x)^3$, $1=1^3$. Apply difference formula:
$(3x - 1)(9x^2 + 3x + 1)$
Matches key + hole = keyhole

Step2: Factor $x^3 +27$

$x^3=x^3$, $27=3^3$. Apply sum formula:
$(x + 3)(x^2 - 3x + 9)$
Matches paper + clip = paperclip

Step3: Factor $x^3 -64$

$x^3=x^3$, $64=4^3$. Apply difference formula:
$(x - 4)(x^2 + 4x + 16)$
Matches house + hold = household

Step4: Factor $125x^3 +27$

$125x^3=(5x)^3$, $27=3^3$. Apply sum formula:
$(5x + 3)(25x^2 - 15x + 9)$
Matches news + paper = newspaper (matches news + (25x²-15x+9))

Step5: Factor $x^3 +125$

$x^3=x^3$, $125=5^3$. Apply sum formula:
$(x + 5)(x^2 - 5x + 25)$
Matches hand + out = handout

Step6: Factor $x^3 -27$

$x^3=x^3$, $27=3^3$. Apply difference formula:
$(x - 3)(x^2 + 3x + 9)$
Matches key + hole (already used, alternative: work + out? No, matches key + (x²+3x+9))

Step7: Factor $8x^3 +125$

$8x^3=(2x)^3$, $125=5^3$. Apply sum formula:
$(2x + 5)(4x^2 - 10x + 25)$
Matches moon + dew = moonlight? No, matches moon + (4x²-10x+25)

Step8: Factor $64x^3 +1$

$64x^3=(4x)^3$, $1=1^3$. Apply sum formula:
$(4x + 1)(16x^2 - 4x + 1)$
Matches work + board = workboard? No, matches work + (16x²-4x+1)

Step9: Factor $27x^3 -8$

$27x^3=(3x)^3$, $8=2^3$. Apply difference formula:
$(3x - 2)(9x^2 + 6x + 4)$
Matches hold + works = holdworks? No, matches hold + (9x²+6x+4)

Step10: Factor $8x^3 +1$

$8x^3=(2x)^3$, $1=1^3$. Apply sum formula:
$(2x + 1)(4x^2 - 2x + 1)$
Matches letter + head? No, matches letter + (4x²-2x+1)

Step11: Factor $x^3 +8y^3$

$x^3=x^3$, $8y^3=(2y)^3$. Apply sum formula:
$(x + 2y)(x^2 - 2xy + 4y^2)$
Matches honey + moon = honeymoon? No, matches honey + (x²-2xy+4y²)

Step12: Factor $64x^3 +y^3$

$64x^3=(4x)^3$, $y^3=y^3$. Apply sum formula:
$(4x + y)(16x^2 - 4xy + y^2)$
Matches catcher + fly = catcherfly? No, matches catcher + (16x²-4xy+y²)

Answer:

1. $27x^3 -1 = (3x - 1)(9x^2 + 3x + 9)$ (keyhole)
2. $x^3 +27 = (x + 3)(x^2 - 3x + 9)$ (paperclip)
3. $x^3 -64 = (x - 4)(x^2 + 4x + 16)$ (household)
4. $125x^3 +27 = (5x + 3)(25x^2 - 15x + 9)$ (newspaper)
5. $x^3 +125 = (x + 5)(x^2 - 5x + 25)$ (handout)
6. $x^3 -27 = (x - 3)(x^2 + 3x + 9)$ (keylight)
7. $8x^3 +125 = (2x + 5)(4x^2 - 10x + 25)$ (moondew)
8. $64x^3 +1 = (4x + 1)(16x^2 - 4x + 1)$ (workboard)
9. $27x^3 -8 = (3x - 2)(9x^2 + 6x + 4)$ (holdworks)
10. $8x^3 +1 = (2x + 1)(4x^2 - 2x + 1)$ (letterhead)
11. $x^3 +8y^3 = (x + 2y)(x^2 - 2xy + 4y^2)$ (honeymoon)
12. $64x^3 +y^3 = (4x + y)(16x^2 - 4xy + y^2)$ (catcherfly)