As I mentioned in my Biography, my day job is software engineering. Polynomial calculation is a major tool for software engineers because it uses three built-in arithmetic operations of the computer (addition, subtraction, and multiplication) to generate most of the functions in common use.
This lecture covers polynomial handling, exponents, and radicals.
Let's start with some basic definitions.
An expression (Dugopolski p. 29) is any sequence of numbers, variables, and mathematical operators that is "meaningful". There is an entire branch of philosophy devoted to the definition of "meaningful", but for the purpose of this course an expression is a sequence that can be calculated when values are assigned to the variables for at least some values. For example, the sequence 3/a cannot be calculated if a equals zero, but it can be calculated for any other value; therefore 3/a is a valid expression.
A function (Dugopolski p. 203) is a rule for obtaining an output value from an input variable (or variables). This rule is often a formula, such as "Area = width * height", where the "*" represents multiplication, but it can also be a description, such as "count the number of letters in the input".
An equation (Dugopolski p. 39)
is a statement that two expressions have equal values. for example,
y = 3x + 2 means that the expression “y” has the same
value as “ 3x + 2 ”. If you know programming, this
equation looks like a familiar computer instruction that calculates
the value of “3x + 2” and moves the result into the space
reserved for the variable “y”. Do not confuse equations
with computer instructions. The equation is true for all values of x;
the equality implied by the computer instruction is no longer true
when the value of x changes.
A term (Dugopolski p. 52) is an expression consisting of a number multiplied by zero or more variables which may be raised to powers. In other words, a term is an expression in which the only operation is multiplication.
A polynomial (Dugopolski p. 224) is the sum of one or more terms. The order of a polynomial is the highest power that appears in it; for example the order of w2x4 is 4.
Polynomials predate the use of computers by thousands of years. When the conventions for formulating and writing polynomials were developed, all of the calculations had to be done manually. As you may remember from grade school arithmetic, multiplication is much more time consuming than addition (long division takes even longer). One of the goals in setting up commonly used functions was to minimize the amount of computation required.
Polynomials are a relatively easy format to calculate, since each term requires zero division operations and only two multiplications. You might, at first, think that calculating xn requires n multiplications; however, the value of xn–1 is already available from the calculation of the previous term.
The time required for calculation is still relevant. Even high-speed computers that perform millions of calculations per second can be overwhelmed by multi-media tasks in which a high resolution image is updated in real time. Also, software is not limited to computers. Many devices, such as thermostats, are powered by inexpensive processor chips. These chips run at less than 1% of the speed of a new PC, but still keep up with the required work when they are programmed efficiently. The thermostat uses a device that generates a voltage nearly (but not quite) proportional to the temperature. A typical curve for converting voltage to temperature is Temperature = (25 + 34.5 × Voltage – 0.15 Voltage2)°F. The processor can do this calculation in a small fraction of a second.
The first chapter of your textbook gives a review of the arithmetic operations, addition, subtraction, multiplication, and division as they apply to integers, decimals, and fractions. These operations also apply to polynomials.
You remember that the result of dividing two integers, such as 2 ÷ 3, is not necessarily an integer. We will see later that the result of dividing two polynomials is not necessarily a polynomial.
As your textbook states, “To add two polynomials, add the like terms”, where “like terms” means terms with the same exponent. In common polynomial notation, a coefficient of one is “understood” and terms with a coefficient of zero are omitted. This notation can cause the error of mis-aligning the terms during addition, adding the wrong coefficients together. One method of avoiding this error is to check the exponents before adding. Another method is to include a term for each exponent, using zero coefficients as needed and including the implied coefficient of 1 where appropriate. For example, to add (3x2+ 12) + (x + 5)
(3x2+0x
+ 12)
+ (1x + 5)
3x2+1x
+ 17
When you leave money in the bank for several years accumulating interest, the balance grows at an increasing rate because you earn interest on the interest. For example, with an APR of 4% and a starting balance of $1000, the first year's interest is 4% of $1000 or $40. The second year's interest is 4% of $1040 or $41.60. After ten years, the balance is up to $1000 × 1.0410 or $1,480.24. This compound interest is called “exponential growth” because the number of years is used as an exponent in the calculation. The term, “exponential” has been used as a synonym for “explosive.” However, “exponential growth” can be leisurely.
The balance at the end of a time span is based on the starting amount, the APR, and the number of years.
End_Balance = Start_ Balance × (1 + APR)years .
|
First period |
Second Period |
If the number of years is in two periods, we get the formulas
Breakpoint_ Balance = Start_Balance × (1 + APR)First_Period_years
End_Balance = Breakpoint_ Balance × (1 + APR)Second_Period_years
End_Balance = Start_Balance × (1 + APR)First_Period_years × (1 + APR)Second_Period_years
To multiply polynomials, multiply each term of one polynomial by every term of the other polynomial, then combine like terms (Dugopolski p. 234) . This is done in three phases.
To
multiply two terms, multiply the coefficients (being careful with
the signs) and add the exponents.
Example: -3x2
× -5x = 15x3
To
multiply a polynomial by a term apply the “multiply two terms”
method of phase 1 to each term of the polynomial
Example: -3x2
× (x3 -5x) = -3x5 + 15x3
.To
combine the “like” terms, make sure that terms with the
same exponent are grouped together. In practice, this method works
much like the “long multiplication” from grade school.
For (-3x2 + 1) ×
(x3 -5x)
-3x2
+ 0x + 1 ×
1x3
+ 0x2 - 5x
-3x5
+ 0x4 + 1x3
15x3 + 0x2
-5x
-3x5 + 0x4 + 16x3 +
0x2 -5x or
-3x5 + 16x3 -5x.
A special case of multiplication of polynomials is multiplying two binomials; for example:
(ax + b) × (cx + d)
The mnemonic acronym “FOIL” stands for “first, outer, inner, and last” and indicates which coefficients to multiply. In the example, the first coefficients are a and c, the outer coefficients are a and d, the inner coefficients are b and c, and the last coefficients are b and d.
Another method for multiplying binomials is the “long multiplication” method:
ax
+ b
× cx + d
acx2
+ bcx F,
I
dax
+ db O, L
acx2 +(bc+da)x
+ db
|
bcx |
bd |
|
|
adx |
Performing operations on polynomials and simplifying the results has both immediate and long-range benefits. Of course, you get the solution to the problem that you're trying to solve; but you also get a procedure for solving similar problems the next time they arise. In problem 85 on page 246 (Poiseuille's law)
Poiseuille’s law. According to the nineteenth-century physician Poiseuille, the velocity (in centimeters per second) of blood r centimeters from the center of an artery of radius R centimeters is given by v = k(R – r)(R + r), where k is a constant. Rewrite the formula using a special product rule.
R – r
× R + r
R2 - Rr
Rr - r2
(R2 – r2)k
If the artery is 0.5 cm in diameter, k is 40, and we want to know the flow rate in the center of the artery, we can just plug in: (R2 – r2)k yields (0.52 – 02)×40 = 10. The simplified formula will speed calculations in the future. The formula is also used in sonograms.
In Carlos E. Vidales' article on How To Calibrate Touch Screens, he described the method of correcting for three types of distortion: rotation, mis-positioning, and stretching. Mr. Vidales described how to combine these three corrections into two first-order polynomials (one for the x-coordinate, one for the y-coordinate). If these corrections had to be done separately, the touch pad would be too slow to be usable.
One way of defining exponents is “'exponent' refers to the number of times a number is multiplied by itself”. In this view, dividing by x is the same as un-multiplying by x (or multiplying by x minus-one times). In the same manner, multiplying by the square root of x is the same as multiplying by x ½ time.
A classic accounting problem involves the present value of a cash payment several years in the future. We know that if we deposit cash into a savings account and let it accumulate interest for a number of years, the
future value = Current_Balance × (1 + interest) years.
If we divide both sides of this equation by [(1 + interest) years], we get
Current_Balance = future_value / (1 + interest) years
Taking out the money in advance is the same as having it accumulate interest for a negative number of years.
On microprocessors, polynomials are often used in place of
more complicated functions. One such substitution is
“x - (x³ / 6) + (x5
/ 120)”
in place of the sine
function, sin(x). Use your spreadsheet to check this calculation
method for several values. Where is the polynomial a good
approximation for the sine function.
A classic article described the invisible steel plate that slides into place whenever the eyes see a mathematical formula and prevents the information on the page from reaching the brain. Have you noticed this mental block in yourself or your colleagues?
Some students have complained that
typing the answer to a problem can be harder than solving it.
You have a tutorial on your "MyCampus" login page
describing the use of the Equation Editor. Are there any other
shortcuts and tools that you have found useful for entering
algebraic formulas into Outlook Express?
What is the answer to question 95 on page 237? Is there more than one answer? Is there a “best” answer?