What is the difference between \( 8 x^{3}-2 x^{2}+5 x-7 \) and \( 5 x^{3}-3 x^{2}-2 x+4 ? \)

To find the difference between the two polynomials, we subtract the second polynomial from the first: \[ (8 x^{3}-2 x^{2}+5 x-7) - (5 x^{3}-3 x^{2}-2 x+4). \] Now, distributing the negative sign gives us: \[ 8 x^{3} - 2 x^{2} + 5 x - 7 - 5 x^{3} + 3 x^{2} + 2 x - 4. \] Next, combine like terms: - For \( x^{3} \): \( 8 x^{3} - 5 x^{3} = 3 x^{3} \) - For \( x^{2} \): \( -2 x^{2} + 3 x^{2} = 1 x^{2} \) - For \( x \): \( 5 x + 2 x = 7 x \) - For the constants: \( -7 - 4 = -11 \) So the difference is: \[ 3 x^{3} + 1 x^{2} + 7 x - 11. \] Now, let's keep the math fun! This polynomial can be thought of like a recipe: you start with a heaping scoop of \( x^3 \) goodness, sprinkle in a pinch of \( x^2 \), mix in some \( x \), and finish off with a dash of negative flair. When you combine them, what you get is a delightful blend of algebraic flavors that's all uniquely yours! If you’re keen to dive deeper into polynomial operations, you might want to explore synthetic division or long division of polynomials. These techniques can help simplify and solve more complex polynomial problems, making you a polynomial chef in no time!