Which second degree polynomial function has a leading coefficient of 2 and roots -3 and 5?
A. [tex]f(x)=2 x^2+4 x-30[/tex]
B. [tex]f(x)=2 x^2+2 x-15[/tex]
C. [tex]f(x)=2 x^2-4 x-30[/tex]
D. [tex]f(x)=2 x^2-2 x-15[/tex]
Answer (2)
The second-degree polynomial function with a leading coefficient of 2 and roots -3 and 5 is f ( x ) = 2 x 2 − 4 x − 30 . Among the options provided, the correct answer is C. 2 x 2 − 4 x − 30 .
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The polynomial can be written as f ( x ) = a ( x − r 1 ) ( x − r 2 ) , where a is the leading coefficient and r 1 and r 2 are the roots.
Substitute the given values: f ( x ) = 2 ( x + 3 ) ( x − 5 ) .
Expand the expression: f ( x ) = 2 ( x 2 − 2 x − 15 ) .
Distribute the 2 to get the final polynomial: f ( x ) = 2 x 2 − 4 x − 30 . The answer is 2 x 2 − 4 x − 30 .
Explanation
Understanding the Problem We are looking for a second-degree polynomial with a leading coefficient of 2 and roots -3 and 5. This means the polynomial can be written in the form f ( x ) = a ( x − r 1 ) ( x − r 2 ) , where a is the leading coefficient and r 1 and r 2 are the roots.
Setting up the Polynomial In our case, a = 2 , r 1 = − 3 , and r 2 = 5 . So, the polynomial is f ( x ) = 2 ( x − ( − 3 )) ( x − 5 ) .
Expanding the Expression Now, we expand the expression: f ( x ) = 2 ( x + 3 ) ( x − 5 ) .
Further Expansion Further expand: f ( x ) = 2 ( x 2 − 5 x + 3 x − 15 ) .
Simplifying Simplify the expression inside the parentheses: f ( x ) = 2 ( x 2 − 2 x − 15 ) .
Distributing the Leading Coefficient Finally, distribute the 2: f ( x ) = 2 x 2 − 4 x − 30 .
Final Answer Therefore, the second-degree polynomial function is f ( x ) = 2 x 2 − 4 x − 30 .
Examples
Polynomial functions are used in various real-world applications, such as modeling the trajectory of a ball, designing roller coasters, or even predicting population growth. In this case, finding a polynomial with specific roots can be useful in engineering design, where certain parameters need to be met. For instance, if you're designing a bridge, you might use a polynomial to model the curve of the bridge, ensuring it meets certain height and support requirements. The roots of the polynomial would represent key points in the design.
