13. [-/1 Points] DETAILS MY NOTES TGEIALG6 5.4.065. Evaluate the polynomial for \( a=-4 \) and \( b=2 \). \( a^{2}+5 a b-b^{2} \) SUBMIT ANSWER

To evaluate the polynomial \( a^{2}+5ab-b^{2} \) for \( a=-4 \) and \( b=2 \), we can substitute these values into the expression. 1. First, calculate \( a^{2} \): \( (-4)^{2} = 16 \). 2. Next, calculate \( 5ab \): \( 5 \cdot (-4) \cdot 2 = -40 \). 3. Then, calculate \( b^{2} \): \( (2)^{2} = 4 \). Now combine these results: \[ 16 - 40 - 4 = 16 - 44 = -28 \]. So, the evaluated result of the polynomial is \(-28\). If you have any queries about polynomials or math in general, exploring the colorful history of polynomial equations can shed light on how these fundamental tools in algebra have evolved and impacted various fields. From ancient civilizations like the Babylonians, who used rudimentary forms of polynomials for trade and land measurement, to modern applications in computer graphics and data analysis, the journey of polynomials is rich and fascinating! For practical applications, polynomials are not just academic constructs; they are central to coding algorithms, physics simulations, and economic forecasts. For instance, in physics, polynomials help model trajectories of projectiles, while in economics, they serve to approximate complex cost functions and predict market behaviors. This versatility makes them invaluable tools across disciplines.