This theorem forms the foundation for solving polynomial equations. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. This is the essence of the Rational Zero Theorem it is a means to give us a pool of possible rational zeros.įigure 1 Using the Fundamental Theorem of Algebra We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4. Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor. The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomialĬonsider a quadratic function with two zeros, x = 2 5 x = 2 5īy the Factor Theorem, these zeros have factors associated with them. But first we need a pool of rational numbers to test. Using the Rational Zero Theorem to Find Rational ZerosĪnother use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. Use the Factor Theorem to find the zeros of f ( x ) = x 3 + 4 x 2 − 4 x − 16 f ( x ) = x 3 + 4 x 2 − 4 x − 16 Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. We can use the Factor Theorem to completely factor a polynomial into the product of n nįactors. In the complex number system will have n n As we will soon see, a polynomial of degree n n This pair of implications is the Factor Theorem. Then the remainder of the Division Algorithm f ( x ) = ( x − k ) q ( x ) + r f ( x ) = ( x − k ) q ( x ) + r Is a zero of f ( x ), f ( x ), then x − k x − k is a factor of f ( x ). Notice, written in this form, x − k x − k is a factor of f ( x ). Recall that the Division Algorithm states that, given a polynomial dividend f ( x ) f ( x ) and a non-zero polynomial divisor d ( x ) d ( x ), there exist unique polynomials q ( x ) q ( x ) and r ( x ) r ( x ) such thatį ( x ) = ( x − k ) q ( x ) + r f ( x ) = ( x − k ) q ( x ) + rĪnd f ( x ) = ( x − k ) q ( x ) + 0 f ( x ) = ( x − k ) q ( x ) + 0 If the polynomial is divided by x – k, x – k, the remainder may be found quickly by evaluating the polynomial function at k, k, that is, f ( k ) f ( k ) Let’s walk through the proof of the theorem. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. In the last section, we learned how to divide polynomials. Evaluating a Polynomial Using the Remainder Theorem In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations. This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. What should the dimensions of the cake pan be? They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. The cake is in the shape of a rectangular solid. The bakery wants the volume of a small sheet cake to be 351 cubic inches.
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