namenerk

2021-12-28

For the following exercises, find the degree and leading coefficient for the given polynomial.
$x\left(4-{x}^{2}\right)\left(2x+1\right)$

Terry Ray

Step 1
Given expression:
$x\left(4-{x}^{2}\right)\left(2x+1\right)$
Step 2
Solution:
$\left(4x-{x}^{3}\right)\left(2x+1\right)$
$2x\left(4x-{x}^{3}\right)+1\left(4x-{x}^{3}\right)$
$8{x}^{2}-2{x}^{4}+4x-{x}^{3}$
$-2{x}^{4}-{x}^{3}+8{x}^{2}+4x$
Thus degree of polynomials is = 4

Virginia Palmer

Step 1
Expression:
$x\left(4-{x}^{2}\right)\left(2x+1\right)$
Step 2
Apply distributive property to expand the parentheses:
$=x\left(8x+4-2{x}^{3}-{x}^{2}\right)$
And again:
$=8{x}^{2}+4x-2{x}^{4}-{x}^{3}$
Rearrange to write in descending order of power:
$=-2{x}^{4}-{x}^{3}+8{x}^{2}+4x$
The highest power of x is 4 so its degree is 4 and its coefficient is -2.
Result:
The function has a coefficient of -2 and its degree is 4.

Vasquez

Step 1
$x\left(4-{x}^{2}\right)\left(2x+1\right)$
We do not need to expand. Just look for the term with the highest possible degree:
$x\left(-{x}^{2}\right)\left(2x\right)=-2{x}^{4}$