Understanding Polynomial Factors and Coefficients

When working with polynomials, it's often necessary to determine the coefficients of a polynomial based on its given factors. In this article, we will explore how to find the coefficients a, b, and c of the polynomial x^3 ax^2 bx c when x - 2, x - 3, and x - 4 are all factors of the polynomial. We will use the polynomial remainder theorem and synthetic division to solve this problem.

The Polynomial Remainder Theorem and Synthetic Division

The polynomial remainder theorem states that if a polynomial P(x) is divided by x - k, the remainder is P(k). In this context, we will use synthetic division to simplify the process of finding the remainder when dividing P(x) by a linear factor x - k.

Given Factors and Conditions

Given the factors x - 2, x - 3, and x - 4, we know that these are factors of the polynomial x^3 ax^2 bx c. This means that when we substitute each of these roots into the polynomial, the polynomial evaluates to zero.

Using Synthetic Division to Find Remainders

First, let's use synthetic division with each of the roots to find the remainder of the polynomial.

Division by (x - 2): Substitute (x 2) into the polynomial x^3 ax^2 bx c. This gives us the equation: [8 4a 2b c 0] Division by (x - 3): Substitute (x 3) into the polynomial x^3 ax^2 bx c. This gives us the equation: [27 9a 3b c 0] Division by (x - 4): Substitute (x 4) into the polynomial x^3 ax^2 bx c. This gives us the equation: [64 16a 4b c 0]

Solving the System of Equations

We now have a system of three linear equations in three variables:

(4a 2b c -8) (9a 3b c -27) (16a 4b c -64)

To solve this system, we can use various methods such as elimination or substitution. For simplicity, let's subtract the first equation from the second and the second from the third to eliminate (c).

Simplifying the System

Subtract the first equation from the second: [(9a 3b c) - (4a 2b c) -27 - (-8)]Which simplifies to:[5a b -19 quad text{(Equation 4)}] Subtract the second equation from the third: [(16a 4b c) - (9a 3b c) -64 - (-27)]Which simplifies to:[7a b -37 quad text{(Equation 5)}]

Solving for (a) and (b)

Now, subtract Equation 4 from Equation 5 to eliminate (b): [(7a b) - (5a b) -37 - (-19)]Which simplifies to:[2a -18 quad Rightarrow quad a -9]

Substitute (a -9) back into Equation 4 to solve for (b): [5(-9) b -19 quad Rightarrow quad -45 b -19 quad Rightarrow quad b 26]

Finally, substitute (a -9) and (b 26) back into the first equation to solve for (c): [4(-9) 2(26) c -8 quad Rightarrow quad -36 52 c -8 quad Rightarrow quad c -24]

Conclusion

The coefficients of the polynomial are (a -9), (b 26), and (c -24). Therefore, the polynomial is:

[x^3 - 9x^2 26x - 24]

Additionally, we can verify this by expanding the product of the factors:

[(x - 2)(x - 3)(x - 4) x^3 - 9x^2 26x - 24]

Alternative Methods to Find Coefficients

Alternatively, you can calculate the product directly:

[(x - 2)(x - 3)(x - 4) x^3 - (2 3 4)x^2 (2cdot3 3cdot4 4cdot2)x - 2cdot3cdot4 x^3 - 9x^2 26x - 24]

By comparing the coefficients of like terms, we find:

[a -9, quad b 26, quad c -24]

The roots of the polynomial1 are (x 2), (x 3), and (x 4).

References

If you further wish to explore polynomial equations and their roots, consider consulting standard references or textbooks on algebra and polynomial equations.