How to Add a Term to Ensure a Root at 3 in a Polynomial

Polynomials form a fundamental part of algebra, and understanding how to manipulate them is crucial for many applications. In this article, we will explore a specific problem: how to add a term to a given polynomial so that 3 becomes a root of the new polynomial.

Problem Statement

Consider the polynomial given by

x^2 - 5x 4

We want to determine what should be added to this polynomial such that when the resulting polynomial is evaluated at x 3, it equals 0. This process involves several steps:

Evaluating the Polynomial at x3

To understand the necessary adjustment, we first evaluate the polynomial at x 3:

$$ (3)^2 - 5(3) 4 9 - 15 4 -2 $$
Thus, when x 3, the polynomial evaluates to -2.

Determining the Required Adjustment

For 3 to be a root of the resulting polynomial, the evaluation at x 3 must equal 0. Since the current value at 3 is -2, we need to add a value to the polynomial to adjust it to 0. Therefore, we need to add 2 to the polynomial to ensure that

$$ (3)^2 - 5(3) 4 2 0 $$
Hence, the term to be added is 2.

Verification

To verify this, let's define the new polynomial as follows:

$$ F(x) x^2 - 5x 4 A $$

where A is the adjustment term we are determining. Specifically, we want

$$ F(3) 0 $$

Substituting x 3 into the new polynomial, we get:

$$ 3^2 - 5(3) 4 A 0 $$

Simplifying the expression:

$$ 9 - 15 4 A 0 $$
$$ -2 A 0 $$

Solving for A, we find:

$$ A 2 $$

This confirms that adding 2 to the polynomial ensures that 3 is a root.

Alternative Approach: Changing the Constant Term

Another way to approach the problem is to consider the polynomial in a different form. Let the polynomial be written as:

$$ f(x) x^2 - 5x C $$

We know that 3 is a root of the polynomial, so let's set up the equation for this root:

$$ 3^2 - 5(3) C 0 $$

Simplifying this, we get:

$$ 9 - 15 C 0 $$
$$ -6 C 0 $$

Solving for C, we find:

$$ C 6 $$

This means the polynomial should be:

$$ x^2 - 5x 6 $$

Thus, we need to add 2 to the original polynomial x^2 - 5x 4 to get x^2 - 5x 6.

Graphical Interpretation

From a graphical perspective, we can visualize the polynomial as a parabola. The x-axis intersections of the parabola represent the roots of the polynomial. By shifting the parabola vertically, we can always make 3 a root. This explains why there can be infinitely many solutions, as we can adjust the polynomial in various ways to achieve this root.

Conclusion

Therefore, the term that needs to be added to the polynomial x^2 - 5x 4 to ensure that 3 is a root of the resulting polynomial is 2.

Answer: 2