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Polynomial Worksheets

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What is a polynomial?

A polynomial is a mathematical expression that has three allowed ingredients - constants, variables and exponents.

1/2 MV2 is a polynomial that has 1/2 as a constant, M and V as variables and, on the V, an exponent of 2. As a group, 1/2 MV2 is called a term. The 'poly' in polynomial refers to the fact that there can be many terms (poly means many).

As long as no variable divides another and all exponents are whole numbers, a polynomial can have any number of terms (but not an infinite number!).

 How are polynomials used in the real world? Many features of the natural world can best be described by polynomials. For instance, the path that a thrown object will follow (baseball, rock, hammer) is a curve called a parabola, and the formula for this curve is a polynomial. The equation to graph it with x and y coordinates is: y = ax2 + bx + c. (a, b, and c are constants that depend on which parabola you want to describe.) The formula in the first section, 1/2 MV2, is used in physics to calculate the energy of moving objects. It says that the kinetic energy is one-half the mass times the velocity (speed) squared. A basic problem in polynomials. A basic problem in polynomials is solving them for a particular variable. For example, if x is 20, what will the polynomial x2 - 5x + 12 equal? Substituting 20 for x gives 202 - (5 x 20) + 12 = 400 - 100 + 12 = 312. Who invented polynomials? Polynomials as a class of formulas came about as the result of adopting mathematical notation. Although problems that we would now call polynomials existed, they were only described in words. Something like, Five barrels of the best wine, 16 barrels of medium quality wine, and 4 barrels of poor wine. In modern mathematical notation, that would be written as: 5x + 16y + 4z, a polynomial. Notation that allowed compact formulas only became popular in the 18th and 19th centuries, with Leonhard Euler (1707 - 1783) credited with many of the symbols we still use today. He first started using letters near the beginning of the alphabet (a,b,c) to stand for constants, and letters near the end (x,y,z) for variables in polynomial equations. So, although people have been investigating polynomials since 300 BC (or earlier) the branch of mathematics we now call polynomials is much younger. An interesting fact about polynomials. When relationships are written as polynomials, they sometimes reveal something that might otherwise be hidden. The equation, Ke = 1/2 MV2 tells us how much energy (Ke) an object in motion has. It says you take the mass of the object (M) and multiply it be how fast the object is moving squared (V2) and by 1/2. We know, instinctively, that if you use a heavier hammer, you can hit something harder. That is an increase in mass. We also know that swinging a hammer faster will also make it hit harder. What might not be so obvious is that an increase in the speed of the hammer means more than an increase in the weight. Because V (velocity, or speed) is squared, an increase in how fast the hammer is moving increases the final energy more than an increase in the weight. This is why a lighter bullet, moving very fast, can do more damage than a heavier bullet that moves more slowly. It is also why bat speed is so important in baseball. Without polynomials to show the relationship, this wouldn't be as easy to see.