Answer:
Option 3
Step-by-step explanation:
Analyzing All Options:
Option 1: Sum of 2 polynomials is always a polynomial;
This isn't always true; Take for instance:
[tex](2x^3 - 3x^2 + 4x + 5) + (-2x^3 + 3x^2 - 4x + 9)[/tex]
[tex]2x^3 - 3x^2 + 4x + 5 -2x^3 + 3x^2 - 4x + 9[/tex]
[tex]-2x^3+2x^3 + 3x^2- 3x^2 + 4x- 4x + 5 + 9[/tex]
[tex]14[/tex]
The result in this case isn't a polynomial
Option 2: The quotient of two polynomials is always a polynomial
This isn't always true; Take for instance:
[tex]\frac{4x^2 + 2x+ 2}{2x^2 + x + 1}[/tex]
[tex]=\frac{2(2x^2 + x + 1)}{2x^2 + x + 1}[/tex]
[tex]=2[/tex]
The result in this case isn't a polynomial
Option 3: The multiplication of two polynomials is always a polynomial
This is always true; Take for instance:
[tex](2x^3 - 3x^2 + 4x + 5) * (-2x^3 + 3x^2 - 4x + 9)[/tex]
Expand this, and you get a polynomial
Option 4: Subtraction of 2 polynomials is always a polynomial;
This isn't always true; Take for instance:
[tex](2x^3 - 3x^2 + 4x + 5) - (2x^3 - 3x^2 + 4x + 9)[/tex]
[tex]2x^3 - 3x^2 + 4x + 5 - 2x^3 + 3x^2 - 4x - 9[/tex]
[tex]- 2x^3 + 2x^3 + 3x^2 - 3x^2 + 4x - 4x+ 5 - 9[/tex]
[tex]=-4[/tex]
This isn't a polynomial
Hence;
Option 3 answers the question
The statement which is true for all polynomials is; When two polynomials are multiplied, the product is always a polynomial.
Discussion:
The sum of or difference between two polynomials may not necessarily be a polynomial. This is so because, the addition or subtraction of the polynomials may yield a function other than a polynomial.
The same for division of polynomials in which case the quotient may not always be a polynomial.
However, when two polynomials are multiplied, the product is always a polynomial.
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https://brainly.com/question/5085952
