# Find the Bounds of the Zeros p(x)=6x^4+19x^3+24x^2+24x+8 p(x)=6×4+19×3+24×2+24x+8
Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.
Largest Degree: 4
Simplify each term.
Cancel the common factor of 6.
Cancel the common factor.
p(x)=6×46+19×36+24×26+24×6+86
Divide x4 by 1.
p(x)=x4+19×36+24×26+24×6+86
p(x)=x4+19×36+24×26+24×6+86
Cancel the common factor of 24 and 6.
Factor 6 out of 24×2.
p(x)=x4+19×36+6(4×2)6+24×6+86
Cancel the common factors.
Factor 6 out of 6.
p(x)=x4+19×36+6(4×2)6(1)+24×6+86
Cancel the common factor.
p(x)=x4+19×36+6(4×2)6⋅1+24×6+86
Rewrite the expression.
p(x)=x4+19×36+4×21+24×6+86
Divide 4×2 by 1.
p(x)=x4+19×36+4×2+24×6+86
p(x)=x4+19×36+4×2+24×6+86
p(x)=x4+19×36+4×2+24×6+86
Cancel the common factor of 24 and 6.
Factor 6 out of 24x.
p(x)=x4+19×36+4×2+6(4x)6+86
Cancel the common factors.
Factor 6 out of 6.
p(x)=x4+19×36+4×2+6(4x)6(1)+86
Cancel the common factor.
p(x)=x4+19×36+4×2+6(4x)6⋅1+86
Rewrite the expression.
p(x)=x4+19×36+4×2+4×1+86
Divide 4x by 1.
p(x)=x4+19×36+4×2+4x+86
p(x)=x4+19×36+4×2+4x+86
p(x)=x4+19×36+4×2+4x+86
Cancel the common factor of 8 and 6.
Factor 2 out of 8.
p(x)=x4+19×36+4×2+4x+2(4)6
Cancel the common factors.
Factor 2 out of 6.
p(x)=x4+19×36+4×2+4x+2⋅42⋅3
Cancel the common factor.
p(x)=x4+19×36+4×2+4x+2⋅42⋅3
Rewrite the expression.
p(x)=x4+19×36+4×2+4x+43
p(x)=x4+19×36+4×2+4x+43
p(x)=x4+19×36+4×2+4x+43
p(x)=x4+19×36+4×2+4x+43
Create a list of the coefficients of the function except the leading coefficient of 1.
196,4,4,43
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=|43|,|196|,|4|,|4|
The maximum value is the largest value in the arranged data set.
b1=|4|
The absolute value is the distance between a number and zero. The distance between 0 and 4 is 4.
b1=4+1
Add 4 and 1.
b1=5
b1=5
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.
196 is approximately 3.16‾ which is positive so remove the absolute value
b2=196+|4|+|4|+|43|
The absolute value is the distance between a number and zero. The distance between 0 and 4 is 4.
b2=196+4+|4|+|43|
The absolute value is the distance between a number and zero. The distance between 0 and 4 is 4.
b2=196+4+4+|43|
43 is approximately 1.3‾ which is positive so remove the absolute value
b2=196+4+4+43
b2=196+4+4+43
Find the common denominator.
Write 4 as a fraction with denominator 1.
b2=196+41+4+43
Multiply 41 by 66.
b2=196+41⋅66+4+43
Multiply 41 and 66.
b2=196+4⋅66+4+43
Write 4 as a fraction with denominator 1.
b2=196+4⋅66+41+43
Multiply 41 by 66.
b2=196+4⋅66+41⋅66+43
Multiply 41 and 66.
b2=196+4⋅66+4⋅66+43
Multiply 43 by 22.
b2=196+4⋅66+4⋅66+43⋅22
Combine.
b2=196+4⋅66+4⋅66+4⋅23⋅2
Reorder the factors of 3⋅2.
b2=196+4⋅66+4⋅66+4⋅22⋅3
Multiply 2 by 3.
b2=196+4⋅66+4⋅66+4⋅26
b2=196+4⋅66+4⋅66+4⋅26
Combine fractions.
Combine fractions with similar denominators.
b2=19+4⋅6+4⋅6+4⋅26
Multiply.
Multiply 4 by 6.
b2=19+24+4⋅6+4⋅26
Multiply 4 by 6.
b2=19+24+24+4⋅26
Multiply 4 by 2.
b2=19+24+24+86
b2=19+24+24+86
b2=19+24+24+86
Simplify the numerator.
Add 19 and 24.
b2=43+24+86
Add 43 and 24.
b2=67+86
Add 67 and 8.
b2=756
b2=756
Cancel the common factor of 75 and 6.
Factor 3 out of 75.
b2=3(25)6
Cancel the common factors.
Factor 3 out of 6.
b2=3⋅253⋅2
Cancel the common factor.
b2=3⋅253⋅2
Rewrite the expression.
b2=252
b2=252
b2=252
Arrange the terms in ascending order.
b2=1,252
The maximum value is the largest value in the arranged data set.
b2=252
b2=252
Take the smaller bound option between b1=5 and b2=252.
Smaller Bound: 5
Every real root on p(x)=6×4+19×3+24×2+24x+8 lies between -5 and 5.
-5 and 5
Find the Bounds of the Zeros p(x)=6x^4+19x^3+24x^2+24x+8   ## Download our App from the store

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