Factor completely.
$-8 x^6-32 x^5+256 x^4$
Answer (2)
The polynomial − 8 x 6 − 32 x 5 + 256 x 4 can be factored by first factoring out the greatest common factor, which is − 8 x 4 . Then, the remaining quadratic x 2 + 4 x − 32 can be factored into ( x + 8 ) ( x − 4 ) . The complete factored form is (-8x^4(x + 8)(x - 4).
;
Factor out the greatest common factor: − 8 x 4 ( x 2 + 4 x − 32 ) .
Factor the quadratic expression: x 2 + 4 x − 32 = ( x + 8 ) ( x − 4 ) .
Combine the factors to get the completely factored form: − 8 x 4 ( x + 8 ) ( x − 4 ) .
The completely factored form of the polynomial is − 8 x 4 ( x + 8 ) ( x − 4 ) .
Explanation
Problem Analysis We are given the polynomial − 8 x 6 − 32 x 5 + 256 x 4 and our goal is to factor it completely.
Factoring out the GCF First, we identify the greatest common factor (GCF) of the terms. The GCF of − 8 x 6 , − 32 x 5 , and 256 x 4 is − 8 x 4 . We factor this out: − 8 x 6 − 32 x 5 + 256 x 4 = − 8 x 4 ( x 2 + 4 x − 32 )
Factoring the Quadratic Now, we need to factor the quadratic expression x 2 + 4 x − 32 . We look for two numbers that multiply to -32 and add to 4. These numbers are 8 and -4. So, we can write the quadratic as: x 2 + 4 x − 32 = ( x + 8 ) ( x − 4 )
Final Factored Form Finally, we substitute the factored quadratic back into the expression: − 8 x 4 ( x 2 + 4 x − 32 ) = − 8 x 4 ( x + 8 ) ( x − 4 ) Thus, the completely factored form of the polynomial is − 8 x 4 ( x + 8 ) ( x − 4 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Imagine you are designing a rectangular garden and you know the area can be represented by the expression − 8 x 6 − 32 x 5 + 256 x 4 . By factoring this expression into − 8 x 4 ( x + 8 ) ( x − 4 ) , you can determine the possible dimensions of the garden in terms of x . This helps in optimizing the layout and resource allocation for the garden.
