Form a polynomial [tex]f(x)[/tex] with real coefficients having the given degree and zeros. Degree 4; zeros: 2-4i; 4 multiplicity 2
Let a represent the leading coefficient. The polynomial is [tex]f(x)=a[/tex] (Type an expression using [tex]x[/tex] as the variable. Use integers or fractions for any numbers in the expression. Simplify your answer.)
Answer (1)
The polynomial has zeros 2 − 4 i , 2 + 4 i , and 4 (with multiplicity 2).
The polynomial can be written as f ( x ) = a ( x − ( 2 − 4 i )) ( x − ( 2 + 4 i )) ( x − 4 ) 2 .
Expanding the factors, we get f ( x ) = a ( x 2 − 4 x + 20 ) ( x 2 − 8 x + 16 ) .
Multiplying the quadratic factors, we obtain the polynomial f ( x ) = a ( x 4 − 12 x 3 + 68 x 2 − 224 x + 320 ) .
Explanation
Understanding the Problem We are given a polynomial f ( x ) of degree 4 with real coefficients. The zeros are 2 − 4 i and 4 (with multiplicity 2). Since the polynomial has real coefficients, if 2 − 4 i is a zero, then its complex conjugate 2 + 4 i is also a zero. The leading coefficient is a . Our objective is to form the polynomial f ( x ) with the given zeros and degree.
Constructing the Polynomial Since the polynomial has degree 4 and zeros 2 − 4 i , 2 + 4 i , and 4 (with multiplicity 2), we can write the polynomial as f ( x ) = a ( x − ( 2 − 4 i )) ( x − ( 2 + 4 i )) ( x − 4 ) 2
Expanding Complex Conjugate Factors Expand the factors ( x − ( 2 − 4 i )) ( x − ( 2 + 4 i )) as follows: ( x − 2 + 4 i ) ( x − 2 − 4 i ) = (( x − 2 ) + 4 i ) (( x − 2 ) − 4 i ) = ( x − 2 ) 2 − ( 4 i ) 2 = ( x 2 − 4 x + 4 ) − ( − 16 ) = x 2 − 4 x + 20
Expanding Repeated Root Factor Expand the factor ( x − 4 ) 2 as follows: ( x − 4 ) 2 = x 2 − 8 x + 16
Multiplying Quadratic Factors Multiply the two quadratic factors: ( x 2 − 4 x + 20 ) ( x 2 − 8 x + 16 ) = x 4 − 8 x 3 + 16 x 2 − 4 x 3 + 32 x 2 − 64 x + 20 x 2 − 160 x + 320 = x 4 − 12 x 3 + 68 x 2 − 224 x + 320
Final Polynomial Therefore, the polynomial is f ( x ) = a ( x 4 − 12 x 3 + 68 x 2 − 224 x + 320 ) Thus, the polynomial is f ( x ) = a ( x 4 − 12 x 3 + 68 x 2 − 224 x + 320 ) .
Examples
Polynomials are used to model curves and relationships in various fields. For instance, engineers use polynomials to design bridges and structures, ensuring stability and optimal performance. Economists use polynomials to analyze cost functions and predict economic trends. In computer graphics, polynomials are essential for creating smooth curves and realistic images. Understanding polynomial functions is crucial for solving real-world problems across many disciplines.
