Write a polynomial [tex]f(x)[/tex] that satisfies the given conditions.
Polynomial of lowest degree with zeros of [tex]-\frac{1}{6}[/tex] (multiplicity 2) and [tex]\frac{1}{4}[/tex] (multiplicity 1) and with [tex]f(0)=3[/tex].
[tex]f(x)=[/tex]
Answer (1)
Find the general form of the polynomial using the given zeros and their multiplicities: f ( x ) = a ( x + 6 1 ) 2 ( x − 4 1 ) .
Use the condition f ( 0 ) = 3 to solve for the constant a , which gives a = − 432 .
Substitute the value of a back into the general form: f ( x ) = − 432 ( x + 6 1 ) 2 ( x − 4 1 ) .
Expand the polynomial to obtain the final form: f ( x ) = − 432 x 3 − 36 x 2 + 12 x + 3 . The final answer is − 432 x 3 − 36 x 2 + 12 x + 3 .
Explanation
Finding the General Form of the Polynomial We are given that the polynomial f ( x ) has zeros at x = − 6 1 with multiplicity 2 and at x = 4 1 with multiplicity 1. This means that ( x + 6 1 ) 2 and ( x − 4 1 ) are factors of f ( x ) . Thus, the polynomial can be written in the form f ( x ) = a ( x + 6 1 ) 2 ( x − 4 1 ) where a is a constant. We are also given that f ( 0 ) = 3 . We can use this information to find the value of a .
Solving for the Constant a Substitute x = 0 into the expression for f ( x ) :
f ( 0 ) = a ( 0 + 6 1 ) 2 ( 0 − 4 1 ) = a ( 36 1 ) ( − 4 1 ) = − 144 a Since f ( 0 ) = 3 , we have 3 = − 144 a Solving for a , we get a = − 432
Expanding the Polynomial Now we substitute a = − 432 back into the expression for f ( x ) :
f ( x ) = − 432 ( x + 6 1 ) 2 ( x − 4 1 ) Expanding the polynomial, we have f ( x ) = − 432 ( x 2 + 3 1 x + 36 1 ) ( x − 4 1 ) f ( x ) = − 432 ( x 3 − 4 1 x 2 + 3 1 x 2 − 12 1 x + 36 1 x − 144 1 ) f ( x ) = − 432 ( x 3 + 12 1 x 2 − 36 1 x − 144 1 ) f ( x ) = − 432 x 3 − 36 x 2 + 12 x + 3
Final Answer Thus, the polynomial is f ( x ) = − 432 x 3 − 36 x 2 + 12 x + 3
Examples
Polynomials are used to model curves and relationships in various fields. For instance, engineers might use polynomials to design the curve of a road or a roller coaster. Economists use polynomials to model cost and revenue functions, helping businesses make informed decisions about pricing and production levels. In computer graphics, polynomials are used to create smooth curves and surfaces for 3D models and animations.
