# Find the Bounds of the Zeros -4x^3+9x^2+3x-6

-4×3+9×2+3x-6
Write the polynomial as a function of x.
f(x)=-4×3+9×2+3x-6
Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.
Largest Degree: 3
Simplify each term.
Cancel the common factor of -4.
Cancel the common factor.
f(x)=-4×3-4+9×2-4+3x-4+-6-4
Divide x3 by 1.
f(x)=x3+9×2-4+3x-4+-6-4
f(x)=x3+9×2-4+3x-4+-6-4
Move the negative in front of the fraction.
f(x)=x3-9×24+3x-4+-6-4
Move the negative in front of the fraction.
f(x)=x3-9×24-3×4+-6-4
Cancel the common factor of -6 and -4.
Factor -2 out of -6.
f(x)=x3-9×24-3×4+-2⋅3-4
Cancel the common factors.
Factor -2 out of -4.
f(x)=x3-9×24-3×4+-2⋅3-2⋅2
Cancel the common factor.
f(x)=x3-9×24-3×4+-2⋅3-2⋅2
Rewrite the expression.
f(x)=x3-9×24-3×4+32
f(x)=x3-9×24-3×4+32
f(x)=x3-9×24-3×4+32
f(x)=x3-9×24-3×4+32
Create a list of the coefficients of the function except the leading coefficient of 1.
-94,-34,32
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=|-34|,|32|,|-94|
The maximum value is the largest value in the arranged data set.
b1=|-94|
-94 is approximately -2.25 which is negative so negate -94 and remove the absolute value
b1=94+1
Write 1 as a fraction with a common denominator.
b1=94+44
Combine the numerators over the common denominator.
b1=9+44
b1=134
b1=134
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.
-94 is approximately -2.25 which is negative so negate -94 and remove the absolute value
b2=94+|-34|+|32|
-34 is approximately -0.75 which is negative so negate -34 and remove the absolute value
b2=94+34+|32|
32 is approximately 1.5 which is positive so remove the absolute value
b2=94+34+32
b2=94+34+32
Combine fractions.
Combine fractions with similar denominators.
b2=9+34+32
Simplify the expression.
b2=124+32
Divide 12 by 4.
b2=3+32
b2=3+32
b2=3+32
To write 3 as a fraction with a common denominator, multiply by 22.
b2=3⋅22+32
Combine 3 and 22.
b2=3⋅22+32
Combine the numerators over the common denominator.
b2=3⋅2+32
Simplify the numerator.
Multiply 3 by 2.
b2=6+32
b2=92
b2=92
Arrange the terms in ascending order.
b2=1,92
The maximum value is the largest value in the arranged data set.
b2=92
b2=92
Take the smaller bound option between b1=134 and b2=92.
Smaller Bound: 134
Every real root on f(x)=-4×3+9×2+3x-6 lies between -134 and 134.
-134 and 134
Find the Bounds of the Zeros -4x^3+9x^2+3x-6