Math Problem Statement
Solution
Let's work through the factoring process for each of the given functions.
a)
Step 1: Try grouping terms.
- Group the first two terms and the last two terms:
Step 2: Factor out the greatest common factor (GCF) from each group.
- From , factor out :
- From , factor out 4:
The expression is now:
This does not factor further easily by grouping, so this is likely the final factored form for now.
b)
Step 1: Use synthetic or trial and error for factoring. Try factoring using possible roots of the cubic equation.
Possible rational roots are factors of -6 (constant term) divided by factors of 1 (leading coefficient of ), i.e., .
Testing :
- Substituting into the function:
So, is a root.
Step 2: Use synthetic division to divide by .
Performing the division of by gives .
Step 3: Factor .
Thus, the fully factored form is:
c)
Step 1: Let , so the equation becomes:
Step 2: Factor the quadratic in .
Step 3: Substitute back .
Step 4: Factor the difference of squares.
Thus, the fully factored form is:
Would you like any further details on these steps or any clarifications?
Additional Practice Questions:
- How do you apply synthetic division to cubic functions?
- What are the common techniques for factoring polynomials by grouping?
- Can you explain the process of factoring the difference of squares?
- How do you determine possible rational roots of a polynomial?
- What are the techniques to factor higher-degree polynomials, such as quartic ones?
Tip:
When factoring polynomials, always check for possible common factors and apply grouping or synthetic division techniques for higher-degree polynomials.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
Factoring by Grouping
Synthetic Division
Difference of Squares
Formulas
Factoring cubic and quartic polynomials
Synthetic division for polynomials
Difference of squares: a^2 - b^2 = (a - b)(a + b)
Theorems
Rational Root Theorem
Factor Theorem
Suitable Grade Level
Grades 9-12
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