Factor polynomial $$ 64x^{6}-1 $$

$$ 64x^{6}-1 = ( 2x-1 )( 4x^{2}+2x+1 )( 2x+1 )( 4x^{2}-2x+1 ) $$

Step 1 :

Rewrite $ 64x^{6}-1 $ as:

$$ 64x^{6}-1 = (8x^{3})^2 - (1)^2 $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = 8x^{3} $ and $ II = 1 $ , we have:

$$ 64x^{6}-1 = (8x^{3})^2 - (1)^2 = ( 8x^{3}-1 ) ( 8x^{3}+1 ) $$

Step 2 :

To factor $ 8x^{3}-1 $ we can use difference of cubes formula:

$$ I^3 - II^3 = (I - II)(I^2 + I \cdot II + II^2) $$

After putting $ I = 2x $ and $ II = 1 $ , we have:

$$ 8x^{3}-1 = ( 2x-1 ) ( 4x^{2}+2x+1 ) $$

Step 3 :

Step 3: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 4 , b = 2 ~ \text{ and } ~ c = 1 $$

Step 4: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 4 $$

Step 5: Find out two numbers that multiply to $ a \cdot c = 4 $ and add to $ b = 2 $.

Step 6: All pairs of numbers with a product of $ 4 $ are:

PRODUCT = 4
1     4	-1     -4
2     2	-2     -2

Step 7: Find out which factor pair sums up to $\color{blue}{ b = 2 }$

Step 8: Because none of these pairs will give us a sum of $ \color{blue}{ 2 }$, we conclude the polynomial cannot be factored.

Step 4 :

To factor $ 8x^{3}+1 $ we can use sum of cubes formula:

$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$

After putting $ I = 2x $ and $ II = 1 $ , we have:

$$ 8x^{3}+1 = ( 2x+1 ) ( 4x^{2}-2x+1 ) $$

Step 5 :

Step 5: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 4 , b = 2 ~ \text{ and } ~ c = 1 $$

Step 6: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 4 $$

Step 7: Find out two numbers that multiply to $ a \cdot c = 4 $ and add to $ b = 2 $.

Step 8: All pairs of numbers with a product of $ 4 $ are:

PRODUCT = 4
1     4	-1     -4
2     2	-2     -2

Step 9: Find out which factor pair sums up to $\color{blue}{ b = 2 }$

Step 10: Because none of these pairs will give us a sum of $ \color{blue}{ 2 }$, we conclude the polynomial cannot be factored.

Step 6 :

Step 6: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 4 , b = -2 ~ \text{ and } ~ c = 1 $$

Step 7: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 4 $$

Step 8: Find out two numbers that multiply to $ a \cdot c = 4 $ and add to $ b = -2 $.

Step 9: All pairs of numbers with a product of $ 4 $ are:

PRODUCT = 4
1     4	-1     -4
2     2	-2     -2

Step 10: Find out which factor pair sums up to $\color{blue}{ b = -2 }$

Step 11: Because none of these pairs will give us a sum of $ \color{blue}{ -2 }$, we conclude the polynomial cannot be factored.

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