Problem 49 - Algebra Form and Function

Question

What is the value of $$5(x-1)(x-2)+2(x-1)(x-3)-4(x-2)(x-3)$$ when \(x=3 ?\)

Step-by-Step Solution

Answer

Answer: The value of the given expression when \(x=3\) is 10.

Step 1: Substitute the value of x

Replace x with 3 in the given expression: $$5(3-1)(3-2)+2(3-1)(3-3)-4(3-2)(3-3)$$

Step 2: Simplify the expressions within the parentheses

Simplify the numbers inside the parentheses: $$5(2)(1)+2(2)(0)-4(1)(0)$$

Step 3: Multiply the numbers

Perform the multiplications: $$10(1)+4(0)-4(0)$$

Step 4: Simplify the sum

Add and subtract the results from the previous step: $$10+0-0$$

Step 5: Obtain the final value

When \(x=3\), the value of the given expression is: $$10$$

Key Concepts

Polynomial Evaluation

Polynomial evaluation is about finding the value of a polynomial expression when a specific variable value is given. It involves substituting the variable with the given number and computing the result. In our exercise, we start with the expression:

• \(5(x-1)(x-2)+2(x-1)(x-3)-4(x-2)(x-3)\)

To evaluate this polynomial, we substitute the variable \(x\) with a specific number, here \(x = 3\). Once replaced, the expression looks like:

• \(5(3-1)(3-2) + 2(3-1)(3-3) - 4(3-2)(3-3)\)

This substitution is a major step as it allows us to transform an algebraic expression into a purely numerical one, which is easier to work with. After substitution, we then move forward to solve the expression to find its numerical value.

Substitution Method

The substitution method simplifies an algebraic expression by replacing variables with numerical values. This approach is like "plugging in" a specific number where the variable appears.
In our example, when we substitute \( x = 3 \), every occurrence of \(x\) in the expression is replaced:

• \(5(\textcolor{blue}{x}-1)(\textcolor{blue}{x}-2)\) becomes \(5(3-1)(3-2)\)
• \(2(x-1)(x-3)\) becomes \(2(3-1)(3-3)\)
• \(-4(x-2)(x-3)\) becomes \(-4(3-2)(3-3)\)

This results in a numeric expression that no longer contains variables. Once the numbers are in place, the task becomes simplifying the expression—a process that involves basic arithmetic operations such as subtraction, multiplication, and addition. This initial substitution step is crucial as it lays down the groundwork for further simplification.

Simplification Process

The simplification process involves breaking down a complex expression into its simplest form. After substitution in our exercise, we are left with the expression:

• \(5(2)(1) + 2(2)(0) - 4(1)(0)\)

Here’s a step-by-step breakdown:

• Evaluate Within Parentheses: Start by calculating expressions within the parentheses. For instance, \((3-1)\) results in \(2\) and \((3-2)\) results in \(1\).
• Perform Multiplication: Multiply the numbers together, e.g., \(5 \times 2 \times 1 = 10\).
• Combine Terms: Add or subtract the results from the multiplication, \(10 + 0 - 0\).

The final result, \(10\), shows the value of the original polynomial expression when \(x=3\). Simplifying helps in understanding and solving expressions more effectively, making complex algebraic problems much more approachable.