Solution:** To evaluate \( g(2) \) for the polynomial \( g(x) = 2x^3 - 3x^2 + 5x - 7 \), substitute \( x = 2 \) into the polynomial:
["Evaluating ( g(2) ) for the Polynomial ( g(x) = 2x^3 - 3x^2 + 5x - 7 ): A Step-by-Step Guide", "When working with polynomials, one essential task is evaluating the function at specific values. In this article, we’ll explore how to evaluate ( g(2) ) for the polynomial ( g(x) = 2x^3 - 3x^2 + 5x - 7 ), using straightforward substitution and arithmetic—concepts vital for mastering algebra and algebraic problem-solving.", "### What Does It Mean to Evaluate a Polynomial?", "Evaluating a polynomial at a specific input value means substituting that value for every instance of ( x ) in the expression. This allows us to simplify and compute the function’s output numerically. For polynomial functions like ( g(x) ), this process follows a clear, ordered sequence of operations that ensure accuracy.", "### The Polynomial in Question", "The given polynomial is:\n[\ng(x) = 2x^3 - 3x^2 + 5x - 7\n]", "We want to compute ( g(2) ), which involves replacing ( x ) with 2 throughout the expression.", "### Step 1: Substitution", "Replace every ( x ) with 2:\n[\ng(2) = 2(2)^3 - 3(2)^2 + 5(2) - 7\n]", "### Step 2: Apply the Exponent Rules", "Evaluate the powers of 2:\n- ( 2^3 = 2 \ imes 2 \ imes 2 = 8 )\n- ( 2^2 = 2 \ imes 2 = 4 )", "Now substitute these values:\n[\ng(2) = 2(8) - 3(4) + 5(2) - 7\n]", "### Step 3: Perform Multiplications", "Multiply each term:\n- ( 2 \ imes 8 = 16 )\n- ( -3 \ imes 4 = -12 )\n- ( 5 \ imes 2 = 10 )", "Now rewrite the expression:\n[\ng(2) = 16 - 12 + 10 - 7\n]", "### Step 4: Perform Addition and Subtraction from Left to Right", "- ( 16 - 12 = 4 )\n- ( 4 + 10 = 14 )\n- ( 14 - 7 = 7 )", "### Final Result", "[\ng(2) = 7\n]", "### Conclusion", "Evaluating ( g(2) ) for the polynomial ( g(x) = 2x^3 - 3x^2 + 5x - 7 ) requires only straightforward substitution, applying exponent rules, and performing successful arithmetic operations in sequence. The result is ( g(2) = 7 ), a number that reflects the polynomial’s output at ( x = 2 ).", "Mastering such evaluations strengthens algebraic fluency and supports more advanced math topics. If you're learning polynomials, practice substituting various values into different polynomials—this builds confidence and accuracy step by step."]
