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102 LearnersLast updated on October 16, 2025
Value of a Polynomial
When we are given a polynomial, when given a polynomial, we can find its value by substituting any value for the variable into the polynomial. This section explains how to evaluate a polynomial.
What are the polynomials?
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) = 2x3 + 3x2 - 4x + 11 is a polynomial.
What is the value of 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) = x2 - 4x + 3.
If x = 2, then
P(2) = 22– 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) = 2x2 + 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
Real-Life Applications of Value of a Polynomial
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) = 5x2 + 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.
Common Mistakes and How to Avoid Them in Value of a Polynomial
It is common for students to make calculation errors while finding the value of a polynomial. Being aware of such mistakes makes problem-solving easier and reduces the chances of mistakes.
Mistake 1
Incorrect substitution of values
It is common for students to misplace values or get confused between signs. Using brackets helps separate variable values, making calculations less confusing.
Mistake 2
Performing operations randomly
When a student does not follow the correct order of operations, it leads to incorrect calculations. Follow the PEMDAS or BODMAS rules to avoid wrong results.
Mistake 3
Working with negative numbers
Specific rules are applied to negative numbers, for instance, the square of a negative number is positive, but its cube is negative. Like (- 2) ² = 4, but (-2)3 = -8. Signs also change when a term is moved to the other side of the equation. Double-checking every step while solving an equation helps avoid errors.
Mistake 4
Substituting only a few terms instead of all
Sometimes students leave a couple of terms and replace only some variables, which gives an incomplete value. Make sure to replace every single variable in the polynomial with the chosen value for the variable to get the value of the polynomial. For example, if x = 7, make sure that all x in p(x) are substituted by 7.
Mistake 5
Forgetting multiplication
Terms like 4x have no multiplication signs between them; this can confuse students. Remember that in such cases, where there is no sign, it means to multiply. So if x = 2, then 4x = 4 × 2 = 8 and not 42.
Solved Examples of Value of a Polynomial
Problem 1
Find the value of P(x) = x2 + 4x + 4, at x = 2
16
Explanation
Substitute x = 2,
P(2) = 22 + 4(2) + 4
P(2) = 4 + 8 + 4
P(2) = 16
Problem 2
Find the value of P(x) = 3x3 − x2 + 6x − 1 at x = 1.
7
Explanation
Substitute the value x = 1 in the given polynomial p(x)
P(1) = 3(1)3 - (1)2 + 6(1) - 1
P(1) = 3 – 1 + 6 – 1
P(1) = 7
Problem 3
Find the value of P(x) = 4x − 2 at x = -3
-10
Explanation
We substitute x = -3 in the given polynomial.
P(-3) = 4(-3) - 2
P(-3) = -12 – 2
P(-3) = - 14
Problem 4
If x = 0, find the value of polynomial P(x) = x2 - 7x + 11
11
Explanation
Substituting x = 0 in P(x) = x2 - 7x + 11
We get,
P(0) = 02 – 7(0) + 11 = 11
Problem 5
If x = - 2, find the value of P(x) = 3x2 + 2x - 5
3
Explanation
P(-2) = 3(-2)2 + 2(-2) - 5
P(-2) = 3(4) - 4 – 5 = 12 – 9 = 3
FAQs on Value of a Polynomial
1.What is meant by the value of a polynomial?
2.What are the 4 types of polynomials?
3.Is 7 a polynomial in math?
4.What is the standard identity of polynomials?
5.What is the formula of a polynomial?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
