In mathematics, polynomials are expressions consisting of variables (such as x and y) and coefficients (such as 2, -3, etc.). They are often used to represent mathematical relationships and are fundamental in algebra. Polynomials can be added, subtracted, multiplied, and divided to solve a variety of problems.
Definition of Polynomial
A polynomial is an expression that can be written in the form:
P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
where an, an-1, …, a2, a1, a0 are constants called coefficients, x is the variable, and n is a non-negative integer called the degree of the polynomial.
Sum of Polynomials
When two polynomials are added together, the result is another polynomial. To add two polynomials, simply combine like terms by adding or subtracting the coefficients of the same degree.
Example Problem:
Consider the following two polynomials:
P(x) = 2x3 – 5x2 + 3x – 7
Q(x) = -x3 + 4x2 – x + 2
To find the sum of these two polynomials, we simply add the coefficients of the corresponding terms:
P(x) + Q(x) = (2 + (-1))x3 + (-5 + 4)x2 + (3 + (-1))x + (-7 + 2)
P(x) + Q(x) = x3 – x2 + 2x – 5
Which Polynomial Represents The Sum Below
Now, let’s consider a specific polynomial sum and determine which polynomial represents it. Given the sum below:
3x2 – 4x + 5
We want to find the polynomial that represents this sum. To do this, we need to consider possible polynomials that could add up to the given sum.
Possible Polynomials
Below are some candidate polynomials that might represent the sum above:
- x2 – 4x + 5
- 3x2 – 4x
- 3x2 – 4x – 5
- 3x2 – 4x + 10
- 2x2 – 4x + 5
Identifying the Polynomial
To determine which polynomial represents the given sum 3x2 – 4x + 5, we need to compare the coefficients of each term in the candidate polynomials with the coefficients of the sum polynomial.
Let’s analyze each candidate polynomial:
Candidate Polynomial 1: x2 – 4x + 5
Coefficients: 1, -4, 5
Comparison:
- Coefficient of x2: 1 ≠ 3 (Not a match)
- Coefficient of x: -4 = -4 (Match)
- Constant term: 5 ≠ 5 (Not a match)
Candidate Polynomial 2: 3x2 – 4x
Coefficients: 3, -4, 0
Comparison:
- Coefficient of x2: 3 = 3 (Match)
- Coefficient of x: -4 = -4 (Match)
- Constant term: 0 ≠ 5 (Not a match)
Candidate Polynomial 3: 3x2 – 4x – 5
Coefficients: 3, -4, -5
Comparison:
- Coefficient of x2: 3 = 3 (Match)
- Coefficient of x: -4 = -4 (Match)
- Constant term: -5 ≠ 5 (Not a match)
Candidate Polynomial 4: 3x2 – 4x + 10
Coefficients: 3, -4, 10
Comparison:
- Coefficient of x2: 3 = 3 (Match)
- Coefficient of x: -4 = -4 (Match)
- Constant term: 10 ≠ 5 (Not a match)
Candidate Polynomial 5: 2x2 – 4x + 5
Coefficients: 2, -4, 5
Comparison:
- Coefficient of x2: 2 ≠ 3 (Not a match)
- Coefficient of x: -4 = -4 (Match)
- Constant term: 5 ≠ 5 (Not a match)
Final Verdict
After analyzing the coefficients of the candidate polynomials, we can conclude that the polynomial that represents the sum 3x2 – 4x + 5 is:
3x2 – 4x
By matching the coefficients of each term correctly, we have identified the polynomial that corresponds to the given sum.
Conclusion
Polynomials are powerful mathematical tools that can be used to represent various mathematical expressions. When adding or subtracting polynomials, it is essential to combine like terms by comparing coefficients and degrees. By carefully analyzing the coefficients of candidate polynomials, we can identify the polynomial that represents a given sum accurately.
Next time you encounter a sum of polynomials, remember to compare coefficients and terms to find the correct polynomial representation. Practice adding and subtracting polynomials to enhance your algebra skills and mathematical understanding.
