Value of a Polynomial

A polynomial is an algebraic expression consisting of terms with coefficients, variables, and their exponents. These exponents are positive whole numbers and do not include negative powers, decimals, or square roots. The terms in a polynomial are separated either by (-) or (+) signs. For instance, \(P(x) = 2x^3 + 3x^2 - 4x + 11\) is a polynomial.

For a polynomial P(x), if x = a then the value of the polynomial P(x) is P(a). Let us take a polynomial \(P(x) = x^2 - 4x + 3\).

If x = 2, then

\(P(2) = 2^2– 4(2) + 3 = 4 – 8 + 3 = -1\)

The value of the polynomial changes depending on the value of x.

How to find the value of a polynomial expression?

The value of a polynomial P(x), its value, can be found by substituting x for a number or constant. 

Let us take a polynomial, \(P(x) = 2x^2 + 3x - 5\)

To find the value of a polynomial, let x = 2,

\(P(2) = 2(2)^2 + 3(2) - 5 = 2(4) + 6 – 5 = 8 + 6 – 5 = 9\)

So, the value of P(x) at x = 2 is 9.

This process can be applied to any value of x.

Let’s take x = 3, then

\(P(3) = 2(3)^2 + 3(3) - 5 = 2(9) + 9–5 = 18 + 4 = 22\)

Mastering the value of polynomials is a key skill in algebra that connects directly to substitution, graphing, and real-world problem-solving. Here are some of the tips and tricks to master the concept of value of polynomials.
 

1. Understand what you’re doing, always remember to replace, then calculate. Think of the process in two steps. The first one being replacing the variable with the given number. Then calculate carefully using order of operations (BODMAS/PEDMAS).
    

2. Always use brackets when substituting, in order to avoid sign mistakes. Every time you substitute a number (especially negatives), wrap it in parentheses.
    

3. Follow the correct order while simplifying:

   Brackets

   Orders (powers/squares)

   Division/multiplication

   Addition/subtraction

   This prevents errors in multistep calculations.
    

4. Check with a calculator, but don’t skip steps. Use calculators to verify, not to replace understanding. Write each step clearly so you can see how the value is built.
    

5. Challenge yourself to spot the fast route. Once you’re comfortable, look for shortcuts. Try to factorize before substituting, if it simplifies the work.

Polynomials are used to predict, calculate, and optimize outcomes that describe patterns and changes. Here are some examples from real life where polynomials are used.
 

• Projectile motion in physics: The height of an object over time is modeled using polynomials; evaluating them gives the height at any point.
   
• Profit and cost functions in economics: Profit and cost are generally modeled as a polynomial. The cost function shows the total cost of producing x items, and the profit function shows the total profit after selling x items. \(C(x) = 5x^2 + 15x + 250\) is an example of a cost function.
   
• Structural design in engineering: Engineers use polynomial expressions to calculate stress, load, or resistance in structures. Substituting specific values in these expressions helps them examine the safety and accuracy in structural designs.
   
• Curve modeling in computer graphics: Shapes like Bézier curves are defined using polynomials. This is especially useful in computer graphics, as evaluating them at certain points helps render smooth curves on the screen.
   
• Crop yield estimation in agriculture: Growth patterns are modeled using polynomial equations. By substituting environmental values such as soil quality, temperature, fertilizers, etc., farmers can predict how much crop they can yield.