# Find the Bounds of the Zeros 4.5x*10^9x^8*1/2

4.5x⋅109×8⋅12
Write the polynomial as a function of x.
f(x)=4.5x⋅(109×8)⋅12
Create a list of the coefficients of the function except the leading coefficient of 1.
4.5,109,12
There will be two bound options, b1 and b2, the smaller of which is the answer. To calculate the first bound option, find the absolute value of the largest coefficient from the list of coefficients. Then add 1.
Arrange the terms in ascending order.
b1=|12|,|4.5|,|109|
The maximum value is the largest value in the arranged data set.
b1=|109|
Simplify each term.
Raise 10 to the power of 9.
b1=|1000000000|+1
The absolute value is the distance between a number and zero. The distance between 0 and 1000000000 is 1000000000.
b1=1000000000+1
b1=1000000000+1
b1=1000000001
b1=1000000001
To calculate the second bound option, sum the absolute values of the coefficients from the list of coefficients. If the sum is greater than 1, use that number. If not, use 1.
Simplify each term.
The absolute value is the distance between a number and zero. The distance between 0 and 4.5 is 4.5.
b2=4.5+|109|+|12|
Raise 10 to the power of 9.
b2=4.5+|1000000000|+|12|
The absolute value is the distance between a number and zero. The distance between 0 and 1000000000 is 1000000000.
b2=4.5+1000000000+|12|
12 is approximately 0.5 which is positive so remove the absolute value
b2=4.5+1000000000+12
b2=4.5+1000000000+12
Find the common denominator.
Write 4.5 as a fraction with denominator 1.
b2=4.51+1000000000+12
Multiply 4.51 by 22.
b2=4.51⋅22+1000000000+12
Multiply 4.51 and 22.
b2=4.5⋅22+1000000000+12
Write 1000000000 as a fraction with denominator 1.
b2=4.5⋅22+10000000001+12
Multiply 10000000001 by 22.
b2=4.5⋅22+10000000001⋅22+12
Multiply 10000000001 and 22.
b2=4.5⋅22+1000000000⋅22+12
b2=4.5⋅22+1000000000⋅22+12
Combine fractions.
Combine fractions with similar denominators.
b2=4.5⋅2+1000000000⋅2+12
Multiply.
Multiply 4.5 by 2.
b2=9+1000000000⋅2+12
Multiply 1000000000 by 2.
b2=9+2000000000+12
b2=9+2000000000+12
b2=9+2000000000+12
Simplify the numerator.
b2=2000000009+12
b2=20000000102
b2=20000000102
Divide 2000000010 by 2.
b2=1000000005
Arrange the terms in ascending order.
b2=1,1000000005
The maximum value is the largest value in the arranged data set.
b2=1000000005
b2=1000000005
Take the smaller bound option between b1=1000000001 and b2=1000000005.
Smaller Bound: 1000000001
Every real root on f(x)=4.5x⋅(109×8)⋅12 lies between -1000000001 and 1000000001.
-1000000001 and 1000000001
Find the Bounds of the Zeros 4.5x*10^9x^8*1/2