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A new bakery offers decorated sheet cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. 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. What should the dimensions of the cake pan be?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem . If the polynomial is divided by the remainder may be found quickly by evaluating the polynomial function at that is, Let’s walk through the proof of the theorem.
Recall that the Division Algorithm states that, given a polynomial dividend and a non-zero polynomial divisor where the degree of is less than or equal to the degree of , there exist unique polynomials and such that
If the divisor, is this takes the form
Since the divisor is linear, the remainder will be a constant, And, if we evaluate this for we have
In other words, is the remainder obtained by dividing by
If a polynomial is divided by then the remainder is the value
Given a polynomial function evaluate at using the Remainder Theorem.
Use the Remainder Theorem to evaluate at
To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by
The remainder is 25. Therefore,
The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.
If is a zero, then the remainder is and or
Notice, written in this form, is a factor of We can conclude if is a zero of then is a factor of
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