P(x) = 30x - 7x - 7x + 2 (a) Prove That (2x + 1) Is A Factor Of P(x) (b) Factorise P(x) Completely. (c)
Understanding how to analyze polynomial expressions is fundamental in algebra. Among the various operations, factoring polynomials and determining their factors are crucial skills that help simplify complex expressions, solve equations, and analyze functions. In this comprehensive guide, we will explore the polynomial P(x) = 30x - 7x - 7x + 2, focusing on proving whether (2x + 1) is a factor, and subsequently, fully factorizing the polynomial.
---
Understanding the Polynomial P(x) = 30x - 7x - 7x + 2
Before diving into the specific parts of the problem, it is essential to understand the structure of P(x).
Simplifying P(x)
Let’s begin by combining like terms:
- The terms involving x are 30x, -7x, and -7x.
- The constant term is +2.
Adding the x-terms:
30x - 7x - 7x = (30x - 7x) - 7x = 23x - 7x = 16x
Therefore, the simplified form of P(x) is:
P(x) = 16x + 2
This simplification makes the problem more straightforward to analyze and solve.
---
(a) Prove That (2x + 1) Is A Factor Of P(x)
To determine whether (2x + 1) is a factor of P(x), we use the Factor Theorem.
Understanding the Factor Theorem
The Factor Theorem states:
- A polynomial P(x) has a factor (ax + b) if and only if P(-b/a) = 0.
In our case:
- The potential factor is (2x + 1).
- The root corresponding to this factor is found by setting 2x + 1 = 0, which gives:
x = -1/2
Step-by-step proof:
- Identify the root associated with the factor: x = -1/2.
- Evaluate P(x) at x = -1/2: P(-1/2).
- If P(-1/2) = 0, then (2x + 1) is a factor of P(x).
Calculating P(-1/2)
Recall, P(x) = 16x + 2, so:
P(-1/2) = 16 (-1/2) + 2 = -8 + 2 = -6
Since P(-1/2) ≠ 0 (it equals -6), the polynomial does not satisfy the condition for (2x + 1) to be a factor.
Conclusion:
- Because P(-1/2) ≠ 0, (2x + 1) is not a factor of P(x).
---
(b) Factorise P(x) Completely
Even though (2x + 1) is not a factor, the polynomial P(x) is already simplified to:
P(x) = 16x + 2
This is a linear polynomial, and its factorization is straightforward.
Factorising Linear Polynomials
Any linear polynomial of the form ax + b can be factored as:
P(x) = d (x + b/d)
where d is a common factor of the coefficients.
Applying this:
- Coefficients are 16 and 2.
- The greatest common divisor (GCD) of 16 and 2 is 2.
Factoring out 2:
P(x) = 2 (8x + 1)
Thus, the complete factorization of P(x) is:
P(x) = 2(8x + 1)
---
Summary and Final Notes
In this comprehensive analysis, we started by simplifying the polynomial P(x) = 30x - 7x - 7x + 2 to its simplest form, P(x) = 16x + 2. We then examined whether the factor (2x + 1) divides P(x). Using the Factor Theorem, we found that P(-1/2) = -6 ≠ 0, which indicates (2x + 1) is not a factor.
Finally, recognizing that P(x) is linear, we factorized it completely by extracting the GCD of its coefficients, resulting in:
P(x) = 2(8x + 1)
This factorization is useful for solving equations involving P(x) or analyzing its roots.
---
Additional Tips for Polynomial Factorization
- Always simplify the polynomial first to make factoring easier.
- Use the Factor Theorem to test potential factors efficiently.
- Find the GCD of coefficients for linear polynomials to factor out common factors.
- For higher-degree polynomials, polynomial division and synthetic division are useful techniques for factorization.
- Remember that not all factors will divide the polynomial; always verify by substitution.
Conclusion
Understanding the process of factorization and the application of the Factor Theorem is essential in algebra. While the specific polynomial P(x) = 30x - 7x - 7x + 2 simplifies neatly to 16x + 2, the techniques discussed are broadly applicable to more complex polynomials. Recognizing whether a particular binomial is a factor allows for efficient solving of polynomial equations and deeper insights into the nature of polynomial functions.
---
Keywords: Polynomial factorization, Factor Theorem, algebra, linear polynomials, algebraic methods, roots, algebraic expressions, mathematical proofs, algebra tutorials
