Question

which operation is not closed for polynomials?
a ((y^2 + 7y - 6) + (6y + 6y^2 + 5))
b (\frac{(x^2 - 15x - 52)}{(x + 8)})
c ((5x + 3)(2x - 1))
d ((x^2 + 4x) + (10x - 2) + (2x^2 - 2))

Explanation:

To determine which operation is not closed for polynomials, we analyze each option:

Step 1: Recall Closure Property

A set is closed under an operation if the result of the operation on elements of the set is also an element of the set. For polynomials, addition, subtraction, and multiplication always result in a polynomial, but division may not.

Step 2: Analyze Option A (Addition)

Adding two polynomials: \((y^2 + 7y - 6) + (6y + 6y^2 + 5)\). Combine like terms: \(y^2 + 6y^2 + 7y + 6y - 6 + 5 = 7y^2 + 13y - 1\), which is a polynomial. So addition is closed.

Step 3: Analyze Option B (Division)

Dividing \(\frac{x^2 - 15x - 52}{x + 8}\). Let's factor the numerator: \(x^2 - 15x - 52 = (x - 19)(x + 8)\)? Wait, no, let's do polynomial long division or factor. Wait, \(x^2 -15x -52\) divided by \(x + 8\): using synthetic division, root at \(x = -8\). Coefficients: 1 | -15 | -52. Bring down 1. Multiply by -8: -8. Add to -15: -23. Multiply by -8: 184. Add to -52: 132. Wait, that's not zero. Wait, maybe I factored wrong. Wait, actually, \(x^2 -15x -52\) divided by \(x + 8\) is \(x - 23 + \frac{132}{x + 8}\), which is not a polynomial (has a rational function term \(\frac{132}{x + 8}\)). So division here does not result in a polynomial.

Step 4: Analyze Option C (Multiplication)

Multiplying \((5x + 3)(2x - 1)\). Use distributive property: \(5x(2x) + 5x(-1) + 3(2x) + 3(-1) = 10x^2 -5x + 6x -3 = 10x^2 + x -3\), a polynomial. So multiplication is closed.

Step 5: Analyze Option D (Addition/Subtraction)

Adding \((x^2 + 4x) + (10x - 2) + (2x^2 - 2)\). Combine like terms: \(x^2 + 2x^2 + 4x + 10x - 2 - 2 = 3x^2 + 14x -4\), a polynomial. So addition is closed.

Answer:

B. \(\frac{(x^2 - 15x - 52)}{(x + 8)}\)