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# Mth141S36n - polynomial Ex If P(x =(x 2(x 3(x 1 then the...

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Sec. 3.6 – Topics of Polynomial Functions (I) INTERMEDIATE VALUE THEOREM : If f(a) > 0 and f(b) < 0 , then there exists a zero value between a and b . This is helpful in approximating zero values, especially if irrational zero values. Ex. f(x) = 3 2 4 7 10 x x x + - - = (x+1)(x-2)(x+5) f(1) = 1 + 4 – 7 – 10 = - 12 f(3) = 27 + 36 – 21 – 10 = 32 so there must be a zero between 1 and 3. (from factored form we see it is at 2!) Note : If f(a) > 0 and f(b) > 0, this doesn’t mean there is NOT a zero between a and b. Also if f(a) < 0 and f(b) < 0 !! Long Division and Synthetic Division Do examples. Factor Theorem : The polynomial x – k is a factor of the polynomial P(x) if and only if P(k) = 0. This means if you know the factors of a polynomial, you should be able to figure out the zeros. AND if you know the zeros, then you should be able to figure out the factors of the
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Unformatted text preview: polynomial. Ex. If P(x) = (x - 2)(x + 3)(x + 1) then the zeros are 2, -3, and -1. Ex. If the zeros of a polynomial are 4, -2, and ½, then the factors are (x – 4), (x + 2), and (2x – 1) An important concept visited again in this section is the relationship between the following three ideas (Example 7): 1) Solving an equation algebraically (to find the value(s) that make it true). 2) Solving the equation graphically (to find the value(s) that make it true) by setting it equal to 0, and finding the x-intercepts of the graph. . 3) Setting the function equal to 0, and finding the value(s) that make it equal to 0. Also referred to as finding the zeros of the function ....
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