Questions: Will the product of (x^2-8) and (4x^2-7) be a polynomial? yes no How do you know? Polynomials are closed under multiplication. Polynomials are not closed under multiplication. Polynomials are closed under addition. Polynomials are not closed under subtraction.

Solution Steps

To determine if the product of \(x^2 - 8\) and \(4x^2 - 7\) is a polynomial, we need to recall that polynomials are closed under multiplication. This means that the product of two polynomials is always a polynomial.

We start with two polynomials: \[ P(x) = x^2 - 8 \] \[ Q(x) = 4x^2 - 7 \]

To find the product of \( P(x) \) and \( Q(x) \), we use polynomial multiplication: \[ (x^2 - 8)(4x^2 - 7) \]

Using the distributive property: \[ (x^2 - 8)(4x^2 - 7) = x^2 \cdot 4x^2 + x^2 \cdot (-7) + (-8) \cdot 4x^2 + (-8) \cdot (-7) \] \[ = 4x^4 - 7x^2 - 32x^2 + 56 \] \[ = 4x^4 - 39x^2 + 56 \]

The resulting expression \( 4x^4 - 39x^2 + 56 \) is a polynomial because it is a sum of terms of the form \( ax^n \) where \( a \) is a constant and \( n \) is a non-negative integer.

Final Answer