and define $ q(x) = p(x+1) $. What is the coefficient of $ x^2 $ in $ q(x) $?

Wed, 06 May 2026 14:21:56 +0000

Understanding $ q(x) = p(x+1) $: Definition, Expansion, and Finding the Coefficient of $ x^2 $ If you’ve encountered a function defined as $ q(x) = p(x+1) $, you’re exploring a fundamental concept in polynomial algebra known as shifting or translation. This transformation shifts the graph of the original polynomial $ p(x) $ horizontally and plays a crucial role in many areas of mathematics, including calculus, differential equations, and approximation theory. ### What is $ q(x) = p(x+1) $? The expression $ q(x) = p(x+1) $ defines a new function $ q $ in terms of the original polynomial $ p $, shifted one unit to the left. To understand this, consider shifting the graph of $ p $ by 1 unit left: each input $ x $ of $ q $ corresponds to $ x+1 $ in $ p $. This transformation does not alter the degree of $ p $ — if $ p $ is a degree-2 polynomial, $ q $ remains degree 2 — but it changes the coefficients through what’s known as polynomial substitution.