Finite Math Examples

Find the Upper and Lower Bounds y=2x+3
Step 1
Find every combination of .
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Step 1.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 2
Apply synthetic division on when .
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Step 2.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
  
Step 2.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
  
Step 2.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
 
Step 2.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 
Step 2.5
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 3
Since and all of the signs in the bottom row of the synthetic division are positive, is an upper bound for the real roots of the function.
Upper Bound:
Step 4
Apply synthetic division on when .
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Step 4.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
  
Step 4.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
  
Step 4.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
 
Step 4.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 
Step 4.5
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 4.6
Simplify the quotient polynomial.
Step 5
Since and all of the signs in the bottom row of the synthetic division are positive, is an upper bound for the real roots of the function.
Upper Bound:
Step 6
Apply synthetic division on when .
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Step 6.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
  
Step 6.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
  
Step 6.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
 
Step 6.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 
Step 6.5
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 7
Since and all of the signs in the bottom row of the synthetic division are positive, is an upper bound for the real roots of the function.
Upper Bound:
Step 8
Apply synthetic division on when .
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Step 8.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
  
Step 8.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
  
Step 8.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
 
Step 8.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 
Step 8.5
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 8.6
Simplify the quotient polynomial.
Step 9
Since and the signs in the bottom row of the synthetic division alternate sign, is a lower bound for the real roots of the function.
Lower Bound:
Step 10
Apply synthetic division on when .
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Step 10.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
  
Step 10.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
  
Step 10.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
 
Step 10.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 
Step 10.5
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 10.6
Simplify the quotient polynomial.
Step 11
Since and all of the signs in the bottom row of the synthetic division are positive, is an upper bound for the real roots of the function.
Upper Bound:
Step 12
Apply synthetic division on when .
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Step 12.1
Place the numbers representing the divisor and the dividend into a division-like configuration.
  
Step 12.2
The first number in the dividend is put into the first position of the result area (below the horizontal line).
  
Step 12.3
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
 
Step 12.4
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
 
Step 12.5
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Step 13
Since and the signs in the bottom row of the synthetic division alternate sign, is a lower bound for the real roots of the function.
Lower Bound:
Step 14
Determine the upper and lower bounds.
Upper Bounds:
Lower Bounds:
Step 15