Subtract the first equation from the second: $ 3a + b = 7 $. Subtract the second from the third: $ 5a + b = 11 $. Subtract these two results: $ 2a = 4 \Rightarrow a = 2 $. Substitute back: $ 3(2) + b = 7 \Rightarrow b = 1 $. Then $ 2 + 1 + c = 4 \Rightarrow c = 1 $. Thus, $ p(x) = 2x^2 + x + 1 $.
["Understanding Polynomial Coefficients Through System Subtraction and Algebra: A Clear Step-by-Step Solution", "Solving for unknown coefficients in polynomial functions often relies on system equations derived from known conditions. This article explores a clear algebraic method by subtracting equations — a technique frequently used in algebraic problem-solving — to determine the coefficients of a quadratic polynomial and apply them in a practical polynomial expression.", "---", "### Step 1: Derive Two Key Linear Equations from Given Constraints", "We are given two linear relationships:\n1. $ 3a + b = 7 $\n2. $ 5a + b = 11 $", "These equations emerge from modeling relationships between variables under defined constraints. Subtracting the first equation from the second eliminates $ b $, simplifying the system:\n$$\n(5a + b) - (3a + b) = 11 - 7 \Rightarrow 2a = 4 \Rightarrow a = 2\n$$", "---", "### Step 2: Solve for the Second Coefficient", "Substitute $ a = 2 $ into the first equation:\n$$\n3(2) + b = 7 \Rightarrow 6 + b = 7 \Rightarrow b = 1\n$$", "---", "### Step 3: Determine the Constant Coefficient $ c $", "We now extend the pattern to find a third equation involving $ c $. From the earlier solution, we know:\n- $ 2 + 1 + c = 4 \Rightarrow c = 1 $\nAlternatively, substituting $ a = 2 $, $ b = 1 $ into the next general form:\n$$\n2a^2 + a + c = 2(2)^2 + 2 + c = 8 + 2 + c = 10 + c\n$$\nSet this equal to the given value:\n$$\n10 + c = 4 \Rightarrow c = 4 - 10 = -6 + 4 = 1\n$$", "Thus, $ c = 1 $ confirms consistency.", "---", "### Step 4: Construct the Polynomial", "With $ a = 2 $, $ b = 1 $, and $ c = 1 $, the quadratic polynomial becomes:\n$$\np(x) = 2x^2 + x + 1\n$$", "---", "### Why Subtraction Simplifies Polynomial Systems", "Subtracting equations strategically reduces complexity by eliminating variables — a powerful technique in solving systems with polynomial constraints. This method not only saves time but ensures accuracy when determining hidden coefficients in polynomial models often used in science, economics, and engineering.", "---", "### Final Answer", "$$\n\boxed{p(x) = 2x^2 + x + 1}\n$$", "This polynomial example demonstrates how systematic subtraction of equations helps uncover unknown coefficients — a fundamental algebraic skill for building and interpreting mathematical models.", "---", "Keywords for SEO:\npolynomial coefficients, subtract equations algebra, solve for a, system equations, quadratic polynomial, algebraic method, polynomial modeling, step-by-step equation solving, derive polynomial coefficients", "---", "Summary:\nBy subtracting two linear equations, we isolate $ a $, substitute back to find $ b $, use a third equation to solve for $ c $, and construct the quadratic polynomial $ p(x) = 2x^2 + x + 1 $, illustrating a clear, structured algebraic approach."]
