HP 35s User Manual

							 
 
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HP 35s  Roots of polynomials 
 
 
 
 
Polynomials 
 
Roots of a polynomial 
 
Using the SOLVE function 
 
Practice solving for roots of polynomials 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Roots of polynomials 
 
hp calculators - 2 - HP 35s  Roots of polynomials - Version 1.0 
Polynomials 
 
A polynomial is an expression containing one or more terms called monomials. These terms contain one or more 
variables multiplied by a constant coefficient. Each of these variables will have a positive exponent. A term may have an 
exponent of zero, in which case it is a constant term. A polynomial will generally not contain negative exponents or 
division by a term or expression containing a variable.  
 
The degree of a polynomial is determined by the largest exponent of a variable within the expression. 
 
Roots of a polynomial are values that when substituted into the expressions variable cause the polynomial’s value to be 
zero. These would correspond to the X-intercept of a polynomial’s graph. Some polynomials do not have roots that are 
real numbers. However, from the fundamental theorem of algebra, every polynomial has at least one root, if the 
allowable values are expanded to include complex numbers. 
 
Roots of a polynomial 
 
The roots of a polynomial are values of X where the value of the function of x (or the value of the polynomial) is equal to 
zero. For example, the polynomial f(x) = X – 2 has a real root at the value +2. The polynomial X2 – 9 has real roots at the 
values of + 3 and – 3. Not every polynomial has roots that are real numbers. For example, the X2 + 4 has no real roots, 
meaning there are no real values for X that will cause X2 + 4 to equal zero. 
 
Using the SOLVE function 
 
The HP 35s has a very powerful root finding capability built into its SOLVE function. As applied in this training aid, the 
SOLVE function, accessed by pressing the ! key, will be used to find roots from user-written programs computing 
the value of a function. This will involve entering a small program, keying in a small equation into the program using a 
variable, indicating to the HP 35s which variable is being considered as the current function, and then solving for the 
value of that variable when the function is equal to zero. The HP 35s knows which variable to solve for by setting the 
value of the function under consideration using the # function. To indicate to the HP 35s that the variable X is to 
be used, press #$. 
 
This training aid cannot begin to illustrate the wide range of applications available using the built-in solver, but it can 
illustrate some of the more common uses.  
Practice solving for roots of polynomials  
Example 1: Solve for the roots of 4X2 – 2X – Y = 12  
Solution: First, rearrange the equation so that the variable Y is isolated. This is necessary to use the SOLVE function 
as we are doing in this training aid. The rearranged equation is Y = 4X2 – 2X – 12. This is a polynomial. 
 
 Were looking for values of X such that 4X2 – 2X – 12 = 0. First, well enter a program that computes the 
value of the function. If a program already exists in program memory with the name of X, then it will need to 
be cleared. This can be done by pressing %& to have the HP 35s display the list of 
programs in the calculator and then press ( to step through the program labels. When the label of the 
program to be deleted is shown in the display, pressing )*will delete that program from the 
calculators memory. Pressing + will then clear the display and allow you to proceed.   
						

							 
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HP 35s  Roots of polynomials 
 
hp calculators - 3 - HP 35s  Roots of polynomials - Version 1.0 
 In RPN or algebraic mode:  ),)-$./
//012$345412$564/
//7/
 
 Figure 1 
 
 To show the checksum and length of this program, press the following in RPN or algebraic mode. Note that 
the symbol &/means to press the right arrow direction of the cursor key. 
/
/In RPN or algebraic mode: %&8  
 Figure 2 
 
 If the checksum of the program just entered does not equal 8BFA, then you have not entered it correctly.  
 
 To clear the checksum display press:  
 
 In RPN or algebraic mode: ),  
 
 Then, to exit the program environment, press: 
 
 In RPN or algebraic mode: ), 
 
 Store an initial guess for X of 10 into the variable X. Then set the function to X and solve for the value of X.  
 
 In RPN or algebraic mode:  69):$#$)!$/
 
 Figure 3 
 
 Since you feel this equation might have a root larger than this, store a new guess for X of 100 into the 
variable X. There is no need to set the function to X (since it has already been done). Then solve for the 
value of X.  
 
 In RPN or algebraic mode:  699):$)!$/
 
 Figure 4 
   
						

							 
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HP 35s  Roots of polynomials 
 
hp calculators - 4 - HP 35s  Roots of polynomials - Version 1.0 
 The same root is returned. This is a good indication (but certainly not foolproof) that there are no roots 
larger than +2 for this equation. 
 
 To see if there is a root less than +2 for this equation, store a new guess for X of –10 into the variable X. 
Then solve for the value of X.  
 
 In RPN or algebraic mode:  69;):$)!$/
 
 Figure 5 
 
Answer: Roots found for the equation are –1.5 and +2. Note that the HP 35s owners manual provides much more 
information about providing initial guesses for the SOLVE feature. 
 
Example 2: Solve for the roots of f(x) = (1/5) x^3 + (4/5) x^2 – (7/5) x - 2  
Solution: Were looking for values of X such that (1/5) x^3 + (4/5) x^2 – (7/5) x – 2 = 0. We expect three roots. 
 
 First, well enter a program that computes the value of the function. If a program already exists in program 
memory with the name of X, then it will need to be cleared. This can be done by pressing 
%& to have the HP 35s display the list of programs in the calculator and then press ( 
to step through the program labels. When the label of the program to be deleted is shown in the display, 
pressing )*will delete that program from the calculators memory. Pressing + will then clear the 
display and allow you to proceed. 
 
 In RPN or algebraic mode:  ),)-$./
/
/6?0
						

							 
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HP 35s  Roots of polynomials 
 
hp calculators - 5 - HP 35s  Roots of polynomials - Version 1.0 
 
 Then, to exit the program environment, press:), 
 
 Store an initial guess for X of 10 into the variable X. Then set the function to X and solve for the value of X.  
 
 In RPN or algebraic mode:  69):$#$)!$/
 
 Figure 8 
 
 Since you feel this equation might have a root larger than this, store a new guess for X of 100 into the 
variable X. There is no need to set the function to X (since it has already been done). Then solve for the 
value of X.  
 
 In RPN or algebraic mode:  699):$)!$/
 
 Figure 9 
 
 The same root is returned. This is a good indication (but certainly not foolproof) that there are no roots 
larger than +2 for this equation. 
 
 To see what the values of roots less than +2 for this equation might be, store a new guess for X of –10 into 
the variable X. Then solve for the value of X.  
 
 In RPN or algebraic mode:  4;):$)!$/
 
 Figure 10 
 
 In RPN or algebraic mode:  69;):$)!$/
 
 Figure 11 
 
Answer: Roots found for the polynomial are 2, -1, and -5. Note that the HP 35s owners manual provides much more 
information about providing initial guesses for the SOLVE feature. 
   
						

							 
 
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HP 35s  Advanced uses of logarithmic functions 
 
 
 
 
Log and antilog functions 
 
Practice using log and antilog functions 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Advanced uses of logarithmic functions 
 
hp calculators - 2 - HP 35s  Advanced uses of logarithmic functions - Version 1.0 
Log and antilog functions 
 
Before calculators like the HP 35s became easily available, logarithms were commonly used to simply multiplication. 
They are still used in many subjects, to represent large numbers, as the results of integration, and even in number 
theory. 
 
The HP 35s has four functions for calculations with logarithms. These are the “common” logarithm of “x”, !, its 
inverse, !#, the “natural” logarithm of “x”, $% and its inverse, $&. 
 
Common logarithms are also called “log to base 10” and the common logarithm of a number “x” is written 
 
    LOG10 x    or just    LOG x 
 
Natural logarithms are also called “log to base e” and the natural logarithm of a number “x” is written 
 
    LOGe x    or   LN x 
 
Logarithms can be calculated to other bases, for example the log to base two of x is written 
 
    LOG2 x 
 
Some problems need the logarithm of a number to a base n, other than 10 or e. On the HP 35s these can be calculated 
using one of the formulae 
 
    LOGn x  =  LOG10 x ÷ LOG10 n 
 
    LNn x  =  LNe x ÷ LNe n 
 
!# and $& are also called “antilogarithms” or “antilogs”. $& is also called the “exponential” function or 
“exp”. Apart from being the inverses of the log functions, they have their own uses. !# is very useful for entering 
powers of 10, especially in programs where the  key can not be used to enter a power that has been calculated. 
$& is used in calculations where exponential growth is involved.  
 
The ( function can be seen as the base “n” antilog function. If 10x is the inverse of log10 x and ex is the inverse of 
loge x, then yx is the inverse of logy x. 
 
Practice using log and antilog functions 
 
Example 1: Find the common logarithm of 2. 
 
Solution: In RPN mode type )!*
 In algebraic mode type !)+*
*
 Figure 1 
 *
Answer: The common logarithm of 2 is very nearly 0.3010.   
						

							 
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HP 35s  Advanced uses of logarithmic functions 
 
hp calculators - 3 - HP 35s  Advanced uses of logarithmic functions - Version 1.0 
Example 2: A rare species of tree has a trunk whose cross-section changes as 1/x with the height x. (Obviously this 
breaks down at ground level and at the tree top.) The cross section for any such tree is given by A/x, where 
A is the cross-section calculated at 1 meter above the ground. What is the volume of the trunk between 1 
meter and 2 meters above ground? 
 
Solution: The volume is obtained by integrating the cross-section along the length, so it is given by the integral: 
 
   Figure 2 
 
 It is possible to evaluate this integral using the HP 35s integration function, but it is much quicker to note 
that the indefinite integral of 1/x is LN x. The result is therefore 
 
  V = A (LN2 – LN1) 
 
 Since LN 1 is 0, this simplifies to  
 
  V=A LN2 
 
 In RPN mode type )$%. In algebraic mode type $%)+.*
 
 No one is likely to measure tree heights to an accuracy of more than three significant digits, so set the  
 HP 35s to display the answer with just 3 digits after the decimal point, by pressing !,-. 
 
 Figure 3 
 
Answer:  Figure 3 shows that the log to base e of 2 is close to 0.693, so the volume is 0.693A cubic meters. 
 
Example 3: What is the log to base 3 of 5? Confirm the result using the ( function. 
 
Solution: Using the equations given above, the log to base 3 of 5 can be calculated as (log10 5)/(log10 3). 
 
 In RPN mode, press:   /!.!0*
 
 In algebraic mode, press:  !/10!.+*
 
 Figure 4 
 
 That this is correct can be confirmed if the following keys are pressed.  
 In RPN mode:*.2(* 
 In algebraic mode: *.($3+*   
						

							 
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HP 35s  Advanced uses of logarithmic functions 
 
hp calculators - 4 - HP 35s  Advanced uses of logarithmic functions - Version 1.0 
 Figure 5 
 
Answer: The log to base 3 of 5 is 1.465 within the current accuracy setting of the calculator, as shown by Figure 5. 
Calculating 3 to this power gives 5.000 which confirms that the correct value for the log had been obtained.  
 
Example 4: An activity of 200 is measured for a standard of Cr51 (with a half-life of 667.20 hours). How much time  
 will have passed when the activity measured in the sample is 170? The formula for half-life computations  
 is shown in Figure 7.  
   Figure 6 
 
Solution: Rearrange the equation to solve for t, as in Figure 8. 
 
   Figure 7 
 
 Now calculate t. In RPN mode: 
 
 4456)+-57+)770$%86/$%0 
 
 In algebraic mode: 
 
 4456)8$%-570)7710$%6/+ 
  
 Figure 8 
 
Answer: 156.435 hours. Figure 8 shows the result in algebraic mode.