Use synthetic or long division to divide \( x^{3}+10 x^{2}+17 x-12 \) by \( x+3 \). What is the new expression? \( x^{2}-7 x-4 \) \( x^{2}+13 x-4 \) \( x^{2}+7 x+4 \) \( x^{2}+7 x-4 \) \( x^{2}+13 x+4 \) \( x^{2}-7 x+4 \)

To tackle this polynomial division using synthetic division, we set up for dividing \( x^{3}+10 x^{2}+17 x-12 \) by \( x+3 \). Using synthetic division, we take the root of \( x+3 \), which is \( -3 \). 1. Write the coefficients: 1 (for \( x^3 \)), 10 (for \( x^2 \)), 17 (for \( x \)), and -12 (constant). 2. Bring down the 1. Multiply \( -3 \) by 1, yielding -3, and add that to 10, resulting in 7. Repeat this for each coefficient. Following through this process, we find \( x^2 + 7x - 4 \) as the quotient. Thus, the new expression we arrive at is \( x^2 + 7x - 4 \). Did you know that polynomial long division is very similar to numerical division? You can think of it as dividing "the big number" (the polynomial) by "the small number" (the divisor) in parts, similar to dividing 123 by 6! It really helps with breaking down complex equations. Also, if you’re curious about the historical evolution of polynomial division, it has its roots in ancient mathematics. The Babylonians had techniques for solving polynomial equations over 3,000 years ago, laying the groundwork for the more systematic methods we use today for functions and algebra!