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Explore the connection between polynomial functions and real numbers, learn the Remainder and Factor Theorems, and practice division. Uncover factors and zeros effortlessly. Test your knowledge with whiteboard problems!
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Plowing Through Sec. 2.4b with Two New Topics: Remainder and Factor Theorems with more Division Practice Homework: p. 374 33-55 odd
“Fundamental Connections” for Polynomial Functions For a polynomial function f and a real number k, the following statements are equivalent: 1. x = k is a solution (or root) of the equation f(x) = 0. 2. k is a zero of the function f. 3. k is an x-intercept of the graph of y = f(x). 4. x – k is a factor of f(x).
And Our Two New Theorems: Theorem: Remainder Theorem If a polynomial f(x) is divided by x – c, then the remainder is r = f(c). Theorem: Factor Theorem • Let f(x) be a polynomial function: • If f(c)=0, then x-c is a factor of f(x) • If x-c is a factor of f(x), then f(c)=0
Using Our New Theorems Find the remainder when the given function is divided by (a) x – 2, (b) x + 1, and (c) x + 4. (a) (b) (c) Because the remainder for part (c) is zero, x + 4 divides evenly into the function. So, x + 4 is a factor of the function, and –4 is an x-intercept of the graph of the function… (we know all this without ever dividing, factoring, or graphing!!!)
Using Our New Theorems Use the Factor Theorem to determine whether the first polynomial is a factor of the second polynomial. Check f(–1): No, x + 1 is not a factor of the second polynomial, because f (–1) = 2
