Solving a Polynomial x^6 – 7x^3 + 12 by Factoring.

In summary, the conversation was about finding the solution set for a problem involving x6 – 7x3 + 12. The person tried using u = x3 to solve for u2-7u+12, but was unsure how to proceed after getting (u - 4)(u - 3). They asked for help with solving this type of problem. By using the zero product property, the solutions were found to be x = ∛4 and x = ∛3.
  • #1
RidiculousName
28
0
Hello, I have been going through the Wisconsin Placement Exam sample test. I'm trying to figure out how to find the solution set for x6 – 7x3 + 12.

I have tried having u = x3 and solving for u2-7u+12, but I'm unsure what to do once I get (u - 4)(u - 3).

Would someone help me figure out how to solve this type of problem?
 
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  • #2
RidiculousName said:
Hello, I have been going through the Wisconsin Placement Exam sample test. I'm trying to figure out how to find the solution set for x6 – 7x3 + 12.

I have tried having u = x3 and solving for u2-7u+12, but I'm unsure what to do once I get (u - 4)(u - 3).

Would someone help me figure out how to solve this type of problem?

if $x^6-7x^3+12 = 0$ ...

$u = x^3 \implies u^2-7u+12 = 0 \implies (u-4)(u-3) = 0$

the zero product property $\implies u = 4$ or $u = 3$

$u = 4 \implies x^3 = 4 \implies x = \sqrt[3]{4}$

$u = 3 \implies x^3 = 3 \implies x = \sqrt[3]{3}$
 
  • #3
skeeter said:
if $x^6-7x^3+12 = 0$ ...

$u = x^3 \implies u^2-7u+12 = 0 \implies (u-4)(u-3) = 0$

the zero product property $\implies u = 4$ or $u = 3$

$u = 4 \implies x^3 = 4 \implies x = \sqrt[3]{4}$

$u = 3 \implies x^3 = 3 \implies x = \sqrt[3]{3}$

Thank you!
 

Related to Solving a Polynomial x^6 – 7x^3 + 12 by Factoring.

1. How do you factor a polynomial?

To factor a polynomial, you need to find its factors, which are numbers or expressions that can be multiplied together to get the original polynomial. One way to factor a polynomial is by using the distributive property, where you break down the polynomial into smaller expressions and find common factors.

2. What is the degree of a polynomial?

The degree of a polynomial is the highest exponent in the polynomial. In this case, the polynomial x^6 – 7x^3 + 12 has a degree of 6, since the highest exponent is 6.

3. How do you solve a polynomial by factoring?

To solve a polynomial by factoring, you need to find the factors of the polynomial and then set them equal to zero. This will give you the solutions or roots of the polynomial, which are the values of x that make the polynomial equal to zero.

4. What is the difference between factoring and solving a polynomial?

Factoring a polynomial means breaking it down into smaller expressions, while solving a polynomial means finding the values of x that make the polynomial equal to zero. Factoring is a step towards solving a polynomial, as it helps identify the possible solutions.

5. Can all polynomials be factored?

No, not all polynomials can be factored. Some polynomials, such as prime polynomials, cannot be broken down into smaller expressions. However, many polynomials can be factored using various methods, such as the distributive property, grouping, and using special patterns like the difference of squares or sum/difference of cubes.

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