zagonek34

2021-12-14

Determine the correct factorization of the polynomial expression given below.
$128{x}^{4}-54x$
A. $2x\left(4x-3\right)\left(16{x}^{2}+24x+9\right)$
B. $2x\left(4x-3\right)\left(16{x}^{2}+12x+9\right)$
C. $2x\left(4x-3\right)\left(16{x}^{2}-12x+9\right)$
D. $\left(4{x}^{2}+9x\right)\left(32x-6x\right)$

Linda Birchfield

Step 1
Solution: Determine the factorization.
$128{x}^{4}-54x$
$128{x}^{4}-54x=2x\left(64{x}^{3}-27\right)$
$=2x\left(64{x}^{3}-27\right)$
$=2x\left({\left(4x\right)}^{3}-{3}^{3}\right)$
$=2x\left({\left(4x\right)}^{3}-{3}^{3}\right)$
$\therefore {a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)$
Step 2
$128{x}^{4}-54x=2x\left(4x-3\right)\left({\left(4x\right)}^{2}+4x×3+{3}^{2}\right)$
$=2x\left(4x-3\right)\left(16{x}^{2}+12x+9\right)$
Hence factor of $\left(128{x}^{4}-54x\right)=?$
$\left(128{x}^{4}-54x\right)=2x\left(4x-3\right)\left(16{x}^{2}+12x+9\right)$
option (B) is Correct.

Donald Cheek

Step 1
We have to determine the correct factorization of the polynomial expression
$128{x}^{4}-54x$
Step 2
$=2x\left(64{x}^{3}-27\right)$
$=2x\left({\left(4x\right)}^{3}-{\left(3\right)}^{3}\right)$
$=2x\left(4x-3\right)\left({\left(4x\right)}^{2}+3\left(4x\right)+{\left(3\right)}^{2}\right)$
$=2x\left(4x-3\right)\left(16{x}^{2}+12x+9\right)$