Can two polynomials be subtracted such that the difference is a binomial? If so, give an example. If not, explain.
Options:
No.
Yes, (3b+3)-(b+a+3)
Yes, (3b^2-3b)-(b+a+3)
Yes, two polynomials can be subtracted such that the difference is a binomial. The example given is:
(3b+3) - (b+a+3)
Simplifying the expression, we get:
3b + 3 - b - a - 3
Combining like terms:
2b - a
The simplified expression, 2b - a, is a binomial.
Yes, two polynomials can be subtracted such that the difference is a binomial. One example is (3b+3)-(b+a+3).
To find the difference, we subtract each term of the second polynomial from the corresponding term of the first polynomial.
(3b+3) - (b+a+3)
= 3b + 3 - b - a - 3
Simplifying, we combine like terms:
= 2b - a
Therefore, the difference of the polynomials (3b+3)-(b+a+3) is a binomial: 2b - a.
Yes, two polynomials can be subtracted such that the difference is a binomial.
To determine whether the given options can be subtracted to yield a binomial, we need to understand the definitions of polynomials and binomials.
A polynomial is an expression consisting of variables, coefficients, and mathematical operations such as addition and subtraction. It can have one or more terms, where each term is the product of a coefficient and one or more variables raised to powers.
A binomial is a polynomial that consists of exactly two terms, typically separated by a subtraction or addition operation.
Now let's evaluate the options:
Option 1: No.
This option states that subtracting two polynomials will not result in a binomial. However, this is not true, as we can indeed subtract two polynomials to obtain a binomial.
Option 2: Yes, (3b+3)-(b+a+3)
Let's simplify this expression:
(3b + 3) - (b + a + 3)
By applying the distributive property and combining like terms, we have:
3b + 3 - b - a - 3
Combine the like terms:
2b - a
The result of this subtraction is a binomial, 2b - a.
Option 3: Yes, (3b^2-3b)-(b+a+3)
Similarly, let's simplify this expression:
(3b^2 - 3b) - (b + a + 3)
Using the distributive property and combining like terms, we get:
3b^2 - 3b - b - a - 3
Combine like terms:
3b^2 - 4b - a - 3
The result of this subtraction is a binomial, 3b^2 - 4b - a - 3.
Therefore, both option 2 and option 3 demonstrate that subtracting polynomials can indeed yield a binomial.