Math Problem Statement
how would you factorise this quartic?
Solution
To factorize the quartic , let's follow these steps:
Step 1: Look for Rational Roots
Using the Rational Root Theorem, the possible roots are the factors of the constant term divided by the leading coefficient (which is ). These factors are:
We test these by substituting into the polynomial.
-
Testing :
-
Testing :
Thus, is a factor.
Step 2: Polynomial Division
Divide by using synthetic or long division.
Long Division Process:
- Divide the leading term: .
- Multiply: .
- Subtract: .
- Repeat for each term:
- ,
- Multiply: ,
- Subtract: ,
- ,
- Multiply: ,
- Subtract: .
Result of division:
Step 3: Factorize the Cubic
Repeat the process to factorize the cubic.
- Test for roots of :
- ,
- ,
- ,
- .
None of the integer roots work, suggesting it may not factor further nicely.
Final Answer:
The factorization is: You could explore further factorization of the cubic using advanced methods (numerical approximations or factoring techniques).
Would you like to go deeper into solving the cubic part or need more clarification?
Related Questions:
- How can synthetic division simplify polynomial division?
- What is the Rational Root Theorem, and how is it applied?
- How can you verify the factorization of a polynomial?
- When is it possible to completely factorize a cubic polynomial?
- How are complex roots identified in polynomial factorization?
Tip:
Always test integer roots first using the Rational Root Theorem before attempting more advanced factorization techniques!
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Factorization
Rational Root Theorem
Polynomial Long Division
Formulas
Rational Root Theorem
Synthetic Division
Theorems
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12
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