In algebra, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Solution sets in polynomials refer to the values of the variables that satisfy the polynomial equation. In this article, we will explore how to find the polynomial that has 5 as its solution set. Let’s dive into the details:
Understanding Solution Sets
A solution set is a collection of values that satisfy a given equation. In the context of polynomials, the solution set is the set of values for which the polynomial equation is equal to zero. For example, if a polynomial equation f(x) = 0 has a solution set of {5}, it means that the value 5 makes the polynomial equation true.
Finding the Polynomial
Given that the solution set of a polynomial is 5, we need to determine the polynomial that equates to zero when the variable x is substituted with 5. To find the polynomial, we can use the concept of synthetic division or direct substitution.
Synthetic Division
- Step 1: Write the coefficients of the polynomial in descending order of the exponents.
- Step 2: Perform synthetic division using the number 5 as the divisor.
- Step 3: The resulting quotient will be the polynomial that has 5 as its solution set.
Direct Substitution
- Step 1: Substitute the value 5 into the variable x in the polynomial equation.
- Step 2: Simplify the expression to find the polynomial that equals zero when x = 5.
Example
Let’s consider an example to find the polynomial that has 5 as its solution set. Suppose we have the equation f(x) = x^2 – 10x + 25. We will verify if x=5 satisfies this equation.
Synthetic Division Method
Following the synthetic division method:
Step 1: Coefficients of f(x) = [1, -10, 25]
Step 2: Performing synthetic division with the divisor 5:
5 | 1 -10 25
5 | 5 -25 0
Step 3: The quotient obtained is x – 5, which is the polynomial that has 5 as its solution set.
Direct Substitution Method
Using the direct substitution method:
Substitute x=5 in the equation f(x) = x^2 – 10x + 25:
f(5) = (5)^2 – 10(5) + 25 = 25 – 50 + 25 = 0
Therefore, the polynomial f(x) = x^2 – 10x + 25 is the polynomial that has 5 as its solution set.
Conclusion
In conclusion, finding the polynomial that has 5 as its solution set involves substituting the value into the polynomial equation and simplifying it to zero. Both synthetic division and direct substitution methods can be used to determine the polynomial corresponding to the given solution set. Understanding solution sets in polynomials is crucial for solving algebraic equations and polynomial functions.
