To find \( T(2) \), we substitute \( x = 2 \) into the polynomial \( T(x) = 2x^3 - 3x^2 + x - 5 \).
["How to Find ( T(2) ): A Simple Polynomial Evaluation", "Understanding how to evaluate a polynomial at a specific value is a fundamental skill in algebra, essential for problems in mathematics, engineering, and data analysis. In this article, we’ll explore how to find ( T(2) ) for the cubic polynomial ( T(x) = 2x^3 - 3x^2 + x - 5 ) step by step.", "---", "### What Is ( T(2) )?", "When we write ( T(2) ), we mean evaluating the polynomial ( T(x) ) at ( x = 2 ). This means substituting every occurrence of ( x ) in the expression with 2, then simplifying the result.", "The polynomial function is:\n[\nT(x) = 2x^3 - 3x^2 + x - 5\n]", "---", "### Step-by-Step Evaluation", "To compute ( T(2) ), substitute ( x = 2 ):", "[\nT(2) = 2(2)^3 - 3(2)^2 + (2) - 5\n]", "Now calculate each term:", "- ( 2^3 = 8 ), so ( 2 \cdot 8 = 16 )\n- ( 2^2 = 4 ), so ( -3 \cdot 4 = -12 )\n- The linear term is simply ( +2 )\n- The constant is ( -5 )", "Putting it all together:", "[\nT(2) = 16 - 12 + 2 - 5\n]", "Now simplify step by step:", "[\n16 - 12 = 4\n]\n[\n4 + 2 = 6\n]\n[\n6 - 5 = 1\n]", "---", "### Final Result", "[\nT(2) = 1\n]", "---", "### Why This Matters", "Evaluating polynomials is crucial in many real-world applications—from calculating trajectories in physics to modeling financial data. Knowing how to substitute and simplify expressions ensures accuracy in solving both simple and complex algebraic problems.", "---", "### Summary", "To find ( T(2) ):", "1. Substitute ( x = 2 ) into the polynomial:\n[\nT(2) = 2(2)^3 - 3(2)^2 + 2 - 5\n]\n2. Compute powers and products:\n[\n= 16 - 12 + 2 - 5\n]\n3. Simplify step by step:\n[\n= 1\n]", "Thus, ( T(2) = 1 ).\nThis straightforward evaluation helps build confidence in mastering polynomial functions.", "---", "Keywords:\nT(2) evaluation, polynomial substitution, evaluate polynomial, algebra tutorial, compute T(x), cubic polynomial, simplifying expressions, mathematical functions, solving polynomials."]
