Finding a third degree polynomial... stumped

In summary, a third degree polynomial with rational coefficients can be found by setting the given zeros and point to the form f(x) = A(x-6)(x^2+1), where A is a constant. After substitution, the value of A is found to be 1/2, resulting in the polynomial f(x) = 1/2(x-6)(x^2+1). This polynomial satisfies the given requirements of having 6 and -i as zeros and passing through the point (2,-10).
  • #1
lashia
3
0
Problem:

Find a third degree polynomial with rational coefficients if two of its zeros
are 6 and – 𝑖 and it passes through the point (2, -10)So far, I have came up with this:
(x-6)(x^2+1) however, instead of passing through (2,-10), it passes through (2,-20)

Anyone know how to come up with a third degree polynomial with 6,-i as its zeros, and passes through (2,-10)? It has given me a headache!

Thank you in advance.
 
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  • #2
lashia said:
Problem:

Find a third degree polynomial with rational coefficients if two of its zeros
are 6 and – 𝑖 and it passes through the point (2, -10)So far, I have came up with this:
(x-6)(x^2+1) however, instead of passing through (2,-10), it passes through (2,-20)

Anyone know how to come up with a third degree polynomial with 6,-i as its zeros, and passes through (2,-10)? It has given me a headache!

Thank you in advance.

you have done right except a small modification that you need to yo do

$f(x) = A(x-6)(x^2+1)$ where A is a constant
putting the values you get A = 2

so $f(x) = 2(x-6)(x^2+1)$
 
Last edited:
  • #3
Thank you so much.. what a relief!
 
  • #4
kaliprasad said:
you have done right except a small modification that you need to yo do

$f(x) = A(x-6)(x^2+1)$ where A is a constant
putting the values you get A = 2

so $f(x) = 2(x-6)(x^2+1)$

It doesn't seem tp pass through (2,-10) :confused:
 
  • #5
Hi lashia,

kaliprasad's value of $A$ should instead be $A = 1/2$. Check now to see that the polynomial passes through $(2,-10)$ with $A = 1/2$.
 
  • #6
lashia said:
It doesn't seem tp pass through (2,-10) :confused:

Let's go back to:

\(\displaystyle f(x)=A(x-6)\left(x^2+1\right)\)

Now, we set:

\(\displaystyle f(2)=A(2-6)\left(2^2+1\right)=-20A=-10\implies A=\frac{1}{2}\)

And so we have:

\(\displaystyle f(x)=\frac{1}{2}(x-6)\left(x^2+1\right)\) :D
 

FAQ: Finding a third degree polynomial... stumped

What is a third degree polynomial?

A third degree polynomial is a mathematical expression that contains a variable raised to the power of 3, along with other constants and coefficients. It is also known as a cubic polynomial.

How do I find the roots of a third degree polynomial?

To find the roots of a third degree polynomial, you can use the rational root theorem, synthetic division, or the cubic formula. The roots are the values of the variable that make the polynomial equal to zero.

What is the process for graphing a third degree polynomial?

To graph a third degree polynomial, you can use the leading coefficient test to determine the end behavior, find the x- and y-intercepts, plot additional points using the behavior and intercepts, and connect the points with a smooth curve.

Can a third degree polynomial have more than three roots?

Yes, a third degree polynomial can have up to three distinct real roots. It may also have complex roots, which come in pairs, making a total of three real or complex roots.

How can I tell if a polynomial is a third degree polynomial?

A polynomial is a third degree polynomial if the highest exponent of the variable is 3. For example, if the polynomial is 3x^3 + 2x^2 + 5x + 1, it is a third degree polynomial.

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