Factor polynomial $$ x^{6}+16x^{3}+64 $$
Answer
Explanation
Step 1 :
Both the first and third terms are perfect squares.
$$ x^6 = \left( \color{blue}{ x^{3} } \right)^2 ~~ \text{and} ~~ 64 = \left( \color{red}{ 8 } \right)^2 $$The middle term ( $ 16x^3 $ ) is two times the product of the terms that are squared.
$$ 16x^3 = 2 \cdot \color{blue}{x^{3}} \cdot \color{red}{8} $$We can conclude that the polynomial $ x^{6}+16x^{3}+64 $ is a perfect square trinomial, so we will use the formula below.
$$ A^2 + 2AB + B^2 = (A + B)^2 $$In this example we have $ \color{blue}{ A = x^{3} } $ and $ \color{red}{ B = 8 } $ so,
$$ x^{6}+16x^{3}+64 = ( \color{blue}{ x^{3} } + \color{red}{ 8 } )^2 $$Step 2 :
To factor $ x^{3}+8 $ we can use sum of cubes formula:
$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$After putting $ I = x $ and $ II = 2 $ , we have:
$$ x^{3}+8 = ( x+2 ) ( x^{2}-2x+4 ) $$Step 3 :
Step 3: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:
$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = 4 }$$Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ 4 } $.
Step 4: Find out pairs of numbers with a product of $\color{red}{ c = 4 }$.
| PRODUCT = 4 | |
| 1 4 | -1 -4 |
| 2 2 | -2 -2 |
Step 5: 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 $ x^{3}+8 $ we can use sum of cubes formula:
$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$After putting $ I = x $ and $ II = 2 $ , we have:
$$ x^{3}+8 = ( x+2 ) ( x^{2}-2x+4 ) $$Step 5 :
Step 5: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:
$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = 4 }$$Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ 4 } $.
Step 6: Find out pairs of numbers with a product of $\color{red}{ c = 4 }$.
| PRODUCT = 4 | |
| 1 4 | -1 -4 |
| 2 2 | -2 -2 |
Step 7: 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 $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:
$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = 4 }$$Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ 4 } $.
Step 7: Find out pairs of numbers with a product of $\color{red}{ c = 4 }$.
| PRODUCT = 4 | |
| 1 4 | -1 -4 |
| 2 2 | -2 -2 |
Step 8: Because none of these pairs will give us a sum of $ \color{blue}{ -2 }$, we conclude the polynomial cannot be factored.