How to Evaluate Polynomials

Indeterminates and constants are found in polynomials, which are algebraic expressions. Polynomials may be thought of as a type of mathematics. They’re used to express numbers in practically every discipline of mathematics, and they’re particularly significant in others, like calculus. 2x + 9 and x2 + 3x + 11 are polynomials, for example.

Definition

 A polynomial is an algebraic expression in which the exponents of all variables must be whole integers. In any polynomial, the exponents of the variables must be non-negative integers. A polynomial contains both constants and variables, however, we cannot divide by a variable in polynomials.

Many formulae include many variables, such as the formula for the surface area of a rectangular prism: 2ab + 2bc + 2ac, where a, b, and c are the three sides’ lengths. We may calculate the value of the surface area by replacing the length values. We may evaluate or combine like terms in polynomials with more than one variable using the same methods as for polynomials with one variable.

• When we evaluate a polynomial for a given value of each variable, we may substitute that value in the equation and obtain its numerical value. In the example below, x = 2, we replace all of the x’s with the number 2 and simplify the statement by following the sequence of operations.

Solve:  7x2 – 3x + 2 for x = −2.

Solution: 7(−2)2 – 3(−2) + 2

Or 7(4) – 3(−2) + 2

Or 28 + 6 + 2

Or 34 + 2

Or 36

Solve: 8c – 7b for b = 4 and c = 5.

Solution: 8(5) – 7(4)

Or 40 – 28

Or 12

Solve: 4x2y – 2xy2 + x – 7 for x = 3 and y = −1.

Solution: 4(32)(−1) – 2(3)(−1)2 + 3 – 7

Or          4(9)(−1) – 2(3)(1) + 3 – 7

Or           −36 – 6 + 3 – 7

Or      −42 + 3 – 7

Or          −39 – 7

Or          −46

If a polynomial includes like terms, they can be combined to simplify the polynomial. The same exact variables are raised to the same exact power incomparable words. If there are many variables, the same rule applies — each variable is raised to the same precise power. Like terms in the polynomial 3xy3z2 + 5xy3z2 + 6x2y3z can be merged. 3xy3z2 and 5xy3z2 contain the same identical variables, x, y, and z, raised to the same exact powers, x, y3, and z2. 8xy3z2 can be obtained by collecting or combining them. While the variables x, y, and z are the same as 6x2y3z, the exponents are different, with x2 instead of x and z instead of z2. As a result, 6x2y3z cannot be used in conjunction with the other phrases. The reduced polynomial is instead expressed with two terms: 8xy3z2 + 6x2y3z.

Solve: 2xy2  – 8x – 3xy2 + 3x.

Solution: 2xy2  – 8x – 3xy2 + 3x

Or          2xy2 – 3xy2  – 8x + 3x

Or          (2 – 3)xy2  + (−8 + 3)x

Or          −1xy2  + (− 5)x

Or          −1xy2  – 5x

Or          2xy2  – 8x – 3xy2 + 3x  = −xy2  – 5x

Solve: 5ba2 + 3a2 + a2b – 4a2 – 2ab2

Solution: 5ba2 + 3a2 + a2b – 4a2 – 2ab2

(5 + 1)a2b + (3 – 4)a2 – 2ab2

6a2b  – a2  – 2ab2

5ba2 + 3a2 + a2b – 4a2 – 2ab2 = 6a2b – a2 – 2ab2

Conclusion:

Polynomials with many variables can be assessed in the same manner that polynomials with only one variable can. We replace the specified values for the variable and do the computation to simplify the polynomial to a numerical value to evaluate any polynomial. To accurately evaluate a polynomial, the order of operations and integer operations must be applied appropriately. Like we can do with polynomials with one variable, we can simplify polynomials with more than one variable by grouping like terms. The same exact variables must be raised to the same exact power in like words. When dealing with several variables, the order in which they are written is irrelevant.