Polynomials, algebraic expressions, multiplication operations, and resulting expressions share an intrinsic relationship. When a polynomial is subjected to multiplication by another polynomial, the product obtained remains a polynomial. This fundamental property stems from the closure property of polynomial multiplication, revealing that the set of polynomials is closed under multiplication. The resulting polynomial inherits the characteristics of its parent polynomials, maintaining its polynomial form and allowing for further operations within the same domain.
Polynomial Times Polynomial Equals Polynomial
When you multiply two polynomials together, you get another polynomial. This is because polynomials are closed under multiplication, meaning that the product of two polynomials is always a polynomial.
The degree of the product polynomial is equal to the sum of the degrees of the two original polynomials. For example, if you multiply a polynomial of degree 3 by a polynomial of degree 2, the product polynomial will be of degree 5.
The coefficients of the product polynomial are determined by multiplying the coefficients of the two original polynomials in a specific way. The following table shows how the coefficients of the product polynomial are calculated for the terms in the first polynomial multiplied by the terms in the second polynomial:
| Term in First Polynomial | Term in Second Polynomial | Coefficient in Product Polynomial |
|---|---|---|
| a | b | ab |
| a | c | ac |
| b | b | b^2 |
| b | c | bc |
For example, if you multiply the polynomial 2x^3 + 3x^2 – 5 by the polynomial x^2 – 2x + 1, the product polynomial will be 2x^5 – x^4 – 10x^3 + 15x^2 – 5x + 5.
Here is a step-by-step guide to multiplying two polynomials:
- Multiply each term in the first polynomial by each term in the second polynomial.
- Combine like terms.
- Simplify the product polynomial.
For example, to multiply the polynomial 2x^3 + 3x^2 – 5 by the polynomial x^2 – 2x + 1, you would follow these steps:
- Multiply each term in the first polynomial by each term in the second polynomial:
2x^3 * x^2 = 2x^5
2x^3 * (-2x) = -4x^4
2x^3 * 1 = 2x^3
3x^2 * x^2 = 3x^4
3x^2 * (-2x) = -6x^3
3x^2 * 1 = 3x^2
-5 * x^2 = -5x^2
-5 * (-2x) = 10x
-5 * 1 = -5
- Combine like terms:
2x^5 - 4x^4 + 2x^3 + 3x^4 - 6x^3 + 3x^2 - 5x^2 + 10x - 5 = 2x^5 - x^4 - 10x^3 + 15x^2 - 5x + 5
- Simplify the product polynomial:
2x^5 - x^4 - 10x^3 + 15x^2 - 5x + 5
Question 1:
Is the product of two polynomials a polynomial?
Answer:
Yes, the product of two polynomials is a polynomial. When two polynomials are multiplied, the coefficients of the terms are multiplied together, and the exponents are added. The resulting polynomial will have a degree equal to the sum of the degrees of the two original polynomials.
Question 2:
Assistant How is the sum of two polynomials related to the original polynomials?
Answer:
The sum of two polynomials is a polynomial that has the same degree as the polynomial with the higher degree. The coefficients of the terms with the same degree are added together, and the terms with different degrees are left unchanged.
Question 3:
Assistant How is the difference between two polynomials related to the original polynomials?
Answer:
The difference between two polynomials is a polynomial that has the same degree as the polynomial with the higher degree. The coefficients of the terms with the same degree are subtracted from each other, and the terms with different degrees are left unchanged.
Well, there you have it! Polynomials and their multiplication—a match made in mathematical heaven. Remember, when you multiply two polynomials, the result will always be another polynomial, a testament to the beauty and order that exists within the world of mathematics. Thanks for sticking with me through this exploration, and be sure to drop by again soon for more mathematical adventures. Until then, keep multiplying those polynomials with confidence!