To determine which of the provided responses demonstrates that polynomials form a closed system under multiplication, we need to ensure that multiplying any two polynomials results in a polynomial.
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(x)(6x • -3): This is an expression involving multiplication of polynomials:
- The term \(6x, \cdot, -3\) is a polynomial (specifically, \(-18x\)), and multiplying it by another polynomial (the variable \(x\)) gives \(-18x^2\), which is also a polynomial.
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3х^2/4 - 1: This expression contains a polynomial \(3x^2/4\) and a constant \(-1\), but it is not presented in a multiplicative context, so it doesn't demonstrate closure under multiplication.
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(х)(1/2х): In this case, you're multiplying two polynomials:
- \(x\) is a polynomial, and \(1/2x\) is also a polynomial (specifically, it can be viewed as \( \frac{1}{2}x^1\)).
- Their product is \((1/2)x^2\), which is a polynomial.
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1/4x^3 • 5: This multiplication involves \(1/4x^3\) (a polynomial) multiplied by \(5\) (also considered a polynomial as it can be seen as \(5x^0\)). The product is \((1/4 \cdot 5)x^3 = \frac{5}{4}x^3\), which is a polynomial.
Based on these evaluations, the responses that clearly show closure under multiplication of polynomials are:
- (x)(6x • -3)
- (х)(1/2х)
- 1/4x^3 • 5
Each of these involves multiplying polynomials to yield another polynomial. The expression 3х^2/4 - 1 does not represent a multiplication operation. Therefore, all three multiplicative expressions (1, 3, and 4) demonstrate that polynomials form a closed system under multiplication.
1 answer