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The substitution method is a straightforward approach used to evaluate polynomials by replacing the variable with a given value. In this context, we have the polynomial: \[8 y^2 - 3 y + 2\]If we’re given a specific value for \( y \), say \( y = 5 \), we begin by substituting this value into the polynomial instead of the variable \( y \). This means replacing every \( y \) in the polynomial with 5:\[8(5)^2 - 3(5) + 2\]This method simplifies the polynomial to an expression containing only numbers, making it easier to solve using basic arithmetic operations.

A quadratic polynomial is a type of polynomial that involves a variable raised to the second power (squared), such as \( y^2 \), but no higher powers. The general form of a quadratic polynomial isgiven by \( ax^2 + bx + c \). In the given problem, our quadratic polynomial is:\[8 y^2 - 3 y + 2\]- The term \( 8 y^2 \) represents the quadratic term, where 8 is the coefficient of \( y^2 \).- The term \( -3 y \) is the linear term, where -3 is the coefficient of \( y \).- The constant term is 2, independent of the variable \( y \).Evaluating a quadratic polynomial involves performing operations on squared terms, linear terms, and constants. Thus, it's very important to follow each mathematical step carefully.

Performing step-by-step calculations makes solving polynomial problems more manageable by breaking the problem into smaller, more understandable parts. Let's go through the steps to evaluate the polynomial \(8 y^2 - 3 y + 2\) for several values of \( y \):1. **Substitute the given value into the polynomial:** When \( y = 5 \), substitute 5 for every \( y \):\[8(5)^2 - 3(5) + 2\].2. **Calculate square and multiplication operations:** Compute \(5^2 = 25\) and then:\[8(25) - 3(5) + 2 = 200 - 15 + 2\].3. **Perform addition and subtraction:** Solve the arithmetic to get:\[200 - 15 = 185\]\[185 + 2 = 187\].So, \(8(5)^2 - 3(5) + 2\) evaluates to 187.For \( y = -2 \):1. **Substitute \(-2\) into the polynomial:**\[8(-2)^2 - 3(-2) + 2\]2. **Compute the square and multiplications:**\[8(4) - 3(-2) + 2 = 32 + 6 + 2\].3. **Perform addition:**\[32 + 6 = 38\]\[38 + 2 = 40\].So, \(8(-2)^2 - 3(-2) + 2\) evaluates to 40.For \( y = 0 \):1. **Substitute 0 into the polynomial:**\[8(0)^2 - 3(0) + 2\]2. **Compute square and multiplications:**\[8(0) - 3(0) + 2 = 0 - 0 + 2\]3. **Simplify:**So, the evaluation is 2.Breaking each problem down into these steps helps clarify the process and makes evaluating polynomials much simpler.