Which is true about the degree of the sum and difference of the polynomials \( 3 x^{5} y-2 x^{3} y^{4}-7 x y^{3} \) and \( -8 x^{5} y+2 x^{3} y^{4} \) \( +x y^{3} \) ? Both the sum and difference have a degree of 6 . Both the sum and difference have a degree of 7 . The sum has a degree of 6 , but the difference has a degree of 7 . The sum has a degree of 7 , but the difference has a degree of 6 .
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To determine the degrees of the sum and difference of the given polynomials, let’s calculate them. The first polynomial is \( 3x^5y - 2x^3y^4 - 7xy^3 \) and the second polynomial is \( -8x^5y + 2x^3y^4 + xy^3 \). When we combine them, the leading term will dictate the degree. The highest degree term in the first polynomial is \( 3x^5y \) (degree 6), and in the second polynomial, it is \( -8x^5y \) (degree 6) as well. Thus, the highest term remains, and they will cancel out when we subtract. The sum will yield a degree of 6. However, their difference will also feature \( 11x^5y \), reinforcing the degree at 6. Hence, both the sum and difference have a degree of 6. That's an exciting find! When dealing with the degrees of polynomials, always focus first on the highest degree term. It's the key indicator! A common mistake is to overlook combining like terms; this can lower the polynomial's degree unexpectedly. So, careful attention can save you from what would be a rude awakening!
