consider the following polynomials. ab^2 +3ab+8a^2 and -5ab^2
Use the two polynomials to illustrate the following:
a. polynomials are closed under addition.
b. polynomials are closed under subtraction.
c. polynomials are closed under multiplication.
(can someone please explain and like evaluate the steps, I want to understand this.)
for multiplication, just expand out (ab^2 +3ab+8a^2)(-5ab^2)
every term in the product will have some combination of powers of a and b, so it will still be a polynomial.
same for addition and subtraction
a. To show that polynomials are closed under addition, we need to add the two given polynomials and verify if the result is also a polynomial.
The given polynomials are:
1. ab^2 + 3ab + 8a^2
2. -5ab^2
To add these two polynomials, we simply combine like terms:
(ab^2 + 3ab + 8a^2) + (-5ab^2)
First, let's rewrite these polynomials in descending order of exponents:
8a^2 + ab^2 + 3ab + (-5ab^2)
Now, let's combine the like terms:
8a^2 + (ab^2 - 5ab^2) + 3ab
The resulting polynomial is:
8a^2 - 4ab^2 + 3ab
By performing the addition of the two given polynomials, we obtain another polynomial (8a^2 - 4ab^2 + 3ab). Therefore, we can conclude that polynomials are closed under addition.
b. To show that polynomials are closed under subtraction, we need to subtract one polynomial from another and verify if the result is also a polynomial.
The given polynomials are:
1. ab^2 + 3ab + 8a^2
2. -5ab^2
To subtract the second polynomial from the first, we subtract each term:
(ab^2 + 3ab + 8a^2) - (-5ab^2)
Again, let's rewrite these polynomials in descending order of exponents:
8a^2 + ab^2 + 3ab - (-5ab^2)
Now, let's subtract the terms:
8a^2 + (ab^2 + 5ab^2) + 3ab
Simplifying further, we obtain:
8a^2 + 6ab^2 + 3ab
By performing the subtraction of the second polynomial from the first, we obtain another polynomial (8a^2 + 6ab^2 + 3ab). Therefore, we can conclude that polynomials are closed under subtraction.
c. To show that polynomials are closed under multiplication, we need to multiply the two given polynomials and verify if the result is also a polynomial.
The given polynomials are:
1. ab^2 + 3ab + 8a^2
2. -5ab^2
To multiply these two polynomials, we distribute each term of the first polynomial across the terms of the second polynomial:
(ab^2 + 3ab + 8a^2) * (-5ab^2)
Expanding this multiplication, we get:
-5a^2b^4 - 15a^2b^3 - 40a^3b^2
The result is another polynomial (-5a^2b^4 - 15a^2b^3 - 40a^3b^2). Therefore, we can conclude that polynomials are closed under multiplication as well.
In summary:
a. Polynomials are closed under addition: (ab^2 + 3ab + 8a^2) + (-5ab^2) = 8a^2 - 4ab^2 + 3ab
b. Polynomials are closed under subtraction: (ab^2 + 3ab + 8a^2) - (-5ab^2) = 8a^2 + 6ab^2 + 3ab
c. Polynomials are closed under multiplication: (ab^2 + 3ab + 8a^2) * (-5ab^2) = -5a^2b^4 - 15a^2b^3 - 40a^3b^2
Sure! Let's go through the steps to demonstrate that polynomials are closed under addition, subtraction, and multiplication using the given polynomials.
a. Polynomials are closed under addition.
To show that polynomials are closed under addition, we need to perform the addition of the two given polynomials and determine if the result is also a polynomial.
The two given polynomials are:
1. ab^2 + 3ab + 8a^2
2. -5ab^2
To add these two polynomials together, we simply combine like terms:
(ab^2 + 3ab + 8a^2) + (-5ab^2)
Combining the like terms inside the parentheses, we get:
ab^2 + 3ab + 8a^2 - 5ab^2
Now, let's group the like terms together:
(ab^2 - 5ab^2) + (3ab) + (8a^2)
Factoring out the common terms, we have:
b^2(a - 5a) + 3ab + 8a^2
Simplifying further, we get:
-4ab^2 + 3ab + 8a^2
This simplified expression is also a polynomial. Therefore, we have demonstrated that polynomials are closed under addition.
b. Polynomials are closed under subtraction.
To show that polynomials are closed under subtraction, we need to perform the subtraction between the two given polynomials and determine if the result is also a polynomial.
The two given polynomials are:
1. ab^2 + 3ab + 8a^2
2. -5ab^2
To subtract the second polynomial from the first, we change the sign of the second polynomial and then perform the addition:
(ab^2 + 3ab + 8a^2) - (-5ab^2)
Changing the sign of the second polynomial, we have:
ab^2 + 3ab + 8a^2 + 5ab^2
Now, let's group the like terms together:
(ab^2 + 5ab^2) + 3ab + 8a^2
Factoring out the common terms, we have:
b^2(a + 5a) + 3ab + 8a^2
Simplifying further, we get:
6ab^2 + 3ab + 8a^2
This simplified expression is also a polynomial. Therefore, we have demonstrated that polynomials are closed under subtraction.
c. Polynomials are closed under multiplication.
To show that polynomials are closed under multiplication, we need to perform the multiplication between the two given polynomials and determine if the result is also a polynomial.
The two given polynomials are:
1. ab^2 + 3ab + 8a^2
2. -5ab^2
To multiply these two polynomials, we apply the distributive property and multiply each term of the first polynomial by each term of the second polynomial:
(ab^2 + 3ab + 8a^2) * (-5ab^2)
Expanding the multiplication, we have:
-5ab^4 - 15a^2b^3 - 40a^3b^2
This expanded form is also a polynomial. Therefore, we have demonstrated that polynomials are closed under multiplication.
In summary, we have shown that polynomials are closed under addition, subtraction, and multiplication using the given polynomials.