Polynomial with Two Unknowns [SOLVED]

In summary, the conversation discusses how to determine a function with two unknowns in a polynomial equation. The solution involves setting the given x-intercepts equal to each other and solving for the unknowns using simultaneous equations. The conversation also briefly mentions solving for three unknowns and translating functions.
  • #1
AFG34
131
0
[SOLVED] polynomial with two unknowns

Homework Statement


The graph of f(x)= 3x^4 + 14x^3 + px^2 + qx + 24 has x-intercepts -4 and 2. Determine the function.

The Attempt at a Solution



I could solve it if there were only one unknown but i don't know how to do it if there are two unknowns.

What i did so far is plug -4 for x and 0 for the output, got an expression = 0
did the same thing for 2
since both equal 0, i set them equal to each other, simplified and got: 2p-q=13.3
Don't know what to do next.
 
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  • #2
Rather then setting them equal to each other. Solve for p with one of your x-intercepts then plug it in your other set with the other x-intercept.
 
  • #3
but there are two unknowns in each expression
 
  • #4
AFG34 said:
but there are two unknowns in each expression
What roco is getting at is that you can create a system of two simultaneous equations thus;

[tex]f(-4) = 0[/tex]

[tex]f(2) = 0[/tex]
 
  • #5
yes i know that, that is what i initially did. So you get 2 equations, both equal to 0, both have q and p in them. But i don't know what to do next.
 
  • #6
AFG34 said:
yes i know that, that is what i initially did. So you get 2 equations, both equal to 0, both have q and p in them. But i don't know what to do next.
Have you never solved simultaneous equations before?
 
  • #7
no, i haven't.
 
  • #8
AFG34 said:
i don't think so.
Okay, in that case if you post the two equations you obtain I shall walk you through the process.
 
  • #9
1) 0 = 16p - 4q - 104
2) 0 = 4p +2q + 184

then i divided both by 2:
1) 0 = 8p - 2q - 52
2) 0 = 2p + q + 92
 
  • #10
Good, so now multiply (2) by 2 and then add the two equations.
 
  • #11
Did you ever learn to solve matrices in Algebra class?
 
  • #12
8p + 132

ok so p = -11, q = -70

thnx
 
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  • #13
rocophysics said:
Did you ever learn to solve matrices in Algebra class?

nope
 
  • #14
AFG34 said:
nope
Well either way, your answer isn't right b/c your answer should have ended up being in terms of a solution.

I'll go step by step. Let me type this up.
 
  • #15
rocophysics said:
Did you ever learn to solve matrices in Algebra class?
If the OP hasn't met simultaneous equations it's pretty safe to say that they haven't been introduced to linear algebra (which IMHO is over-kill for a question such as this).
AFG34 said:
8p + 132

ok so p = -11
Firstly, what you have written is an expression, not an equation. Secondly, you might want to check your coefficent of p.
 
  • #16
0 = 12p + 132 is an equation
ya i typed it wrong
checked the back of the book, got the right answer (f(x)= 3x^4 + 14x^3 - 11x^2 - 70x + 24). ok i know how to do them now, thnx

hootenanny, funny name
 
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  • #17
AFG34 said:
0 = 12p + 132 is an equation
ya i typed it wrong
checked the back of the book, got the right answer. ok i know how to do them now, thnx
Want to learn the fast way?
 
  • #18
sure

got a quick question:
is the graph of y = log3^(x+4) the same as the graph of y = log3^x+4?
*the base is 3 not 10

I think the second one is y=log3^x moved up by 4 units.
 
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  • #19
This will only apply to linear equations.

[tex]a_1 x+b_1 y=c_1[/tex]
[tex]a_2 x+b_2 y=c_2[/tex]

[tex]x=\frac{\left|\begin{array}{cc}c_1 & b_1 \\ c_2 & b_2\end{array}\right|}{\left|\begin{array}{cc}a_1 & b_1 \\ a_2 & b_2\end{array}\right|}=\frac{c_1 b_2 - b_1 c_2}{a_1 b_2 - b_1 a_2}[/tex]

[tex]y=\frac{\left|\begin{array}{cc}a_1 & c_1 \\ a_2 & c_2\end{array}\right|}{\left|\begin{array}{cc}a_1 & b_1 \\ a_2 & b_2\end{array}\right|}=\frac{a_1 c_2 - c_1 a_2}{a_1 b_2 - b_1 a_2}[/tex]

Notice that the denominator for both x & y are the same (determinant). While the numerators vary.

You can also solve for 3 unknowns but that takes too long to type.
 
Last edited:
  • #20
AFG34 said:
sure

got a quick question:
is the graph of y = log3^(x+4) the same as the graph of y = log3^x+4?
*the base is 3 not 10

I think the second one is y=log3^x moved up by 4 units.
I'm confused to as what your problem is. Is it ...

[tex]y=\log_3{(x+4)}[/tex]

?
 
  • #21
ya that and the second one is the same except there is no bracket around x+4
 
  • #22
AFG34 said:
ya that and the second one is the same except there is no bracket around x+4
[tex]y=\log_3{(x+4)}[/tex]

[tex]y=\log_3 x+4[/tex]

No they are not the same

2nd one, as you said it translated up 4 units. While the 1st one is translated to the left 4 units.

[tex]y=f(x+c)[/tex] translation to the left c units.

[tex]y=f(x)+c[/tex] translation up c units.
 
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  • #23
thnx very much :D
 

1. What is a polynomial with two unknowns?

A polynomial with two unknowns is an algebraic expression that contains two variables, usually represented as x and y. It is a mathematical expression that can be simplified and evaluated for different values of the variables.

2. How do you solve a polynomial with two unknowns?

To solve a polynomial with two unknowns, you need to use a system of equations. This involves setting up two equations with the two unknown variables and solving for both variables simultaneously. You can use various methods such as substitution or elimination to solve the equations.

3. Can a polynomial with two unknowns have more than two terms?

Yes, a polynomial with two unknowns can have any number of terms. The number of terms in a polynomial is determined by the number of different variables and their exponents. For example, the polynomial 2x^2 + 3xy + 4y^2 has three terms.

4. What is the degree of a polynomial with two unknowns?

The degree of a polynomial with two unknowns is the highest sum of exponents in any term of the polynomial. For example, the polynomial 2x^2 + 3xy + 4y^2 has a degree of 3, as the term 2x^2 has the highest sum of exponents (2 + 0 = 2).

5. What are some real-life applications of polynomials with two unknowns?

Polynomials with two unknowns are used in various fields such as physics, engineering, and economics. They can be used to model and solve problems related to real-world scenarios, such as finding the optimal production level for a company or predicting the trajectory of a projectile.

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