Solve for "a" in Polynomial Function: 2x^3 - ax^2 - 12x - 7

In summary, the value of "a" for which the function 2x^3 - ax^2 - 12x - 7 has a repeated factor is 3. This can be found by setting the derivative of the function equal to 0 and solving for "a", which gives the roots of the function as 2 and -1. By plugging in "a=3" into the original equation, we find that -1 is a double root, leading to the factored form 2(x+1)^2(x-7/2).
  • #1
josephcollins
59
0
Could someone please help me with this question, I'd be most obliged:

Q) Find the value of "a" for which the function
2x^3 - ax^2 - 12x - 7
has a repeated factor.

Thanks
 
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  • #2
If you mean two identical roots, then ...

write [tex] 2x^3 - ax^2 - 12x - 7 = (x-b)^2(2x-c) [/tex]

Expand the RHS and compare coefficients. You have 3 equations in 3 unknowns.
 
  • #3
Since we are seeking only a specific solution, we can keep in mind that if the cubic has a double root, that this root is also present in its derative. The derative is: 2(3X^2-aX-6). Here if we let a =3, we get X =2, -1. -1 works in the original equation and the factoring then is 2(x+l)^2(x-7/2).
 

1. What is a polynomial function?

A polynomial function is a mathematical expression that contains variables and constants, and only involves addition, subtraction, and multiplication. It can be written in the form of ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are constants and n is a non-negative integer.

2. How do you solve for "a" in a polynomial function?

To solve for "a" in a polynomial function, we need to use the principles of algebra and rearrange the terms of the function to isolate the variable "a" on one side of the equation. This can be done by grouping like terms and using various algebraic operations to simplify the equation.

3. Can the value of "a" in a polynomial function be negative?

Yes, the value of "a" in a polynomial function can be negative. In fact, the coefficient of the highest degree term (in this case, 2x^3) can be any real number, including negative numbers.

4. Is there a specific method for solving for "a" in a polynomial function?

There are several methods for solving for "a" in a polynomial function, but the most common method is to use the quadratic formula. This formula can be used to solve for any variable in a quadratic equation, including "a" in a polynomial function with a degree of 3 or higher.

5. Can a polynomial function have more than one value for "a"?

No, a polynomial function can only have one value for "a". This value is determined by the specific equation and cannot be changed. However, different polynomial functions may have different values for "a" depending on their coefficients and degree.

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