Write a polynomial [tex]f(x)[/tex] that satisfies the given conditions.
Polynomial of lowest degree with zeros of -4 (multiplicity 3), 1 (multiplicity 1), and with [tex]f(0)=320[/tex].
Answer (1)
f ( x ) = − 5 x 4 − 55 x 3 − 180 x 2 − 80 x + 320
Explanation
Finding the General Form of the Polynomial We are given that the polynomial f ( x ) has zeros at -4 with multiplicity 3 and 1 with multiplicity 1. This means that ( x + 4 ) 3 and ( x − 1 ) are factors of f ( x ) . Since we want the polynomial of lowest degree, we can write the general form of the polynomial as f ( x ) = a ( x + 4 ) 3 ( x − 1 ) , where a is a constant.
Using the Given Condition We are also given that f ( 0 ) = 320 . We can use this condition to find the value of the constant a . Substituting x = 0 into the general form, we get:
f ( 0 ) = a ( 0 + 4 ) 3 ( 0 − 1 ) = a ( 4 3 ) ( − 1 ) = − 64 a
Since f ( 0 ) = 320 , we have − 64 a = 320 .
Solving for the Constant Solving for a , we get:
a = − 64 320 = − 5
So, the polynomial is f ( x ) = − 5 ( x + 4 ) 3 ( x − 1 ) .
Expanding the Polynomial Now, we expand the polynomial to get the final form:
f ( x ) = − 5 ( x + 4 ) 3 ( x − 1 ) = − 5 ( x 3 + 12 x 2 + 48 x + 64 ) ( x − 1 )
f ( x ) = − 5 ( x 4 − x 3 + 12 x 3 − 12 x 2 + 48 x 2 − 48 x + 64 x − 64 )
f ( x ) = − 5 ( x 4 + 11 x 3 + 36 x 2 + 16 x − 64 )
f ( x ) = − 5 x 4 − 55 x 3 − 180 x 2 − 80 x + 320
Final Answer Therefore, the polynomial that satisfies the given conditions is:
f ( x ) = − 5 x 4 − 55 x 3 − 180 x 2 − 80 x + 320
Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a projectile, the shape of a suspension bridge, or the growth of a population. In this case, we constructed a polynomial with specific zeros and a given value at a particular point. This technique can be applied in engineering to design systems with desired characteristics or in physics to model the behavior of particles under certain conditions. For example, if you know certain points where a curve should intersect the x-axis and one other point on the curve, you can determine the polynomial equation that describes the curve.
