Working_with_Polynomials

Working_with_Polynomials - MTHSC 360 Working with...

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MTHSC 360 – Working with Polynomials January 15, 2008 1 Three ways to compute a straight line. In High School Algebra, we learned that the equation for a straight line was given by the expression y = mx + β (1) where m is the slope (the rise over the run) of the straight line and β is the y -intercept. This is called the slope-intercept form for the equation of a straight line. This is a particularly convenient form if we know both the y -intercept, (0 ,y 0 ), and the x -intercept, ( x 0 , 0) because β = y 0 and m = - β/x 0 . Then in Calculus we began to make heavy use of the point-slope form for the equation of a straight line. y = m ( x - a ) + f ( a ) (2) First, given two points ( a,f ( a )) and ( b,f ( b )), m is the slope of the secant line, (the straight line through the two points). m = f ( b ) - f ( a ) b - a Then in the limit as b got closer and closer to a , we got the equation for the tangent line. y = f 0 ( a )( x - a ) + f ( a ) where m = f 0 ( a ). It’s important to note that if the function, f ( x ), is in fact a straight line, then we have m = f 0 ( a ) = f ( b ) - f ( a ) b - a and β = bf ( a ) - af ( b ) b - a However, the big advantage of the point-slope form is that it focuses our attention on the neighborhood of the point ( a,f ( a )) and emphasizes how things change with x – especially for x near a . In other words, when x is close to a , then f ( x ) is close to f ( a ) and the exact value of f ( x ) is obtained by adding the small correction m ( x - a ). Of course we know from geometry that two points determine a straight line, and this leads to a third form of the equation of a straight line given two points, ( a,f ( a )) and ( b,f ( b )). This form is called the two-point form. y = f ( a ) x - b a - b + f ( b ) x - a b - a (3) Although this may seem like a strange way to write an equation for a straight line, it is easy to see that ( x - b ) / ( a - b ) is a polynomial of degree 1 which has the value 1 at x = a and
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the value 0 at x = b . Similarly, ( x - a ) / ( b - a ) is also a polynomial of degrees 1 but it has the value 0 at x = a and the value 1 at x = b . Consequently when we combine these two polynomials by multiplying the first by f ( a ) and the second by f ( b ) and adding the result,
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Working_with_Polynomials - MTHSC 360 Working with...

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