To determine if polynomials form a closed system under multiplication, we need to check if the result of multiplying any two polynomials results in another polynomial.
Let's analyze the expressions you've provided:
- \(5x - 1.2\) is a polynomial of degree 1 (linear).
- \(3\) is a constant polynomial (degree 0).
- \(7 - 2\) simplifies to \(5\), which is also a constant polynomial (degree 0).
- \(5\) is another constant polynomial (degree 0).
- \(2x^3 + x^2\) is a polynomial of degree 3.
To check for closure under multiplication, let's perform some multiplications:
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Multiplying \(5x - 1.2\) and \(2x^3 + x^2\): \[ (5x - 1.2)(2x^3 + x^2) = 10x^4 + 5x^3 - 2.4x^3 - 1.2x^2 = 10x^4 + (5 - 2.4)x^3 - 1.2x^2 \] This result is clearly a polynomial (degree 4).
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Multiplying \(3\) and \(2x^3 + x^2\): \[ 3(2x^3 + x^2) = 6x^3 + 3x^2 \] This result is also a polynomial (degree 3).
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Multiplying \(5\) and \(3\): \[ 5 \cdot 3 = 15 \] This is a constant polynomial (degree 0).
Based on these examples, we see that multiplying two polynomials gives another polynomial each time.
Thus, polynomials are closed under multiplication.
If you are looking for a response that confirms this, consider stating:
"Polynomials form a closed system under multiplication because the result of multiplying two polynomials is also a polynomial."
However, if the options in your question represent responses concerning this closure property, please clarify or provide the possible answers more clearly for further analysis.
1 answer