The word polynomial is made of two words, "poly" and "nomial", meaning many terms. It is used in math for representing expressions. Thus, a polynomial expression is a sentence with a minimum of two numbers and at least one math operation in it.
A polynomial is made up of terms and each term has a coefficient, while an expression is a sentence with a minimum of two numbers and at least one math operation in it. The expressions which satisfy the criterion of a polynomial are polynomial expressions. The following examples to check if they are polynomial expressions or not.
In the two cases discussed above, the expression x^{2} + 3√x + 1 is not a polynomial expression because the variable has a fractional exponent, i.e., 1/2 which is a noninteger value; while for the second expression x^{2} + √3 x + 1, the fractional power 1/2 is on the constant which is 3 in this case, hence it is a polynomial expression.
The standard form of any polynomial expression is given when the terms of expression are ordered from the highest degree to the lowest degree. The polynomial standard form can be written as:
a_{n}X^{n} + a_{n1} X^{n1} + .... + a_{2}X^{2} + a_{1}x + a_{0}. For example, ax^{2} + bx + c.
There are three types of polynomials based on the number of terms that they have:
For a Single Variable Polynomial
We find the degree of a polynomial expression using the following steps:
The highest exponent of the expression gives the degree of the polynomial. Let's consider the polynomial expression, 5x^{3} + 4x^{2}  x^{4}  2x^{3}  5x^{2} + x^{4}. In this case, the expression can be simplified as, 3x^{3}  x^{2}. Here, the highest exponent corresponding to the polynomial expression is 3. Hence, the degree of polynomial expression is 3. Observe the following expression.
The above expression shows:
If we take a polynomial expression with two variables, say x and y, x^{3} + 3x^{2}y^{4} + 4y^{2} + 6. We follow the above steps, with an additional step of adding the powers of different variables in the given terms. and thus the degree of the polynomial is 6. This is because, in 3x^{2}y^{4}, the exponent values of x and y are 2 and 4, respectively. When we add these, we get 6. Hence, the degree of the multivariable polynomial expression is 6. Observe the following polynomial which shows how its degree is considered to be 9.
We can simplify polynomial expressions in the following ways:
By combining like terms
The terms having the same variables are combined using arithmetic operations so that the calculation gets easier. For example, let us simplify the polynomial expression: 5x^{5} + 7x^{3} + 8x + 9x^{3}  4x^{4 } 10x  3x^{5}. We combine the like terms to get, 5x^{5}  3x^{5}  4x^{4} + 7x^{3 }+ 9x^{3 }+ 8x  10x . On simplifying we get, 2x^{5}  4x^{4} + 16x^{3}  2x
The FOIL (First, Outer, Inner, Last) technique is used for the arithmetic operation of multiplication. Each step uses the distributive property. 'First' means multiply the terms which come first in each binomial. Then, 'outer' means multiply the outermost terms in the product, followed by the 'inner' terms, and then the 'last' terms are multiplied. For example, to simplify the given polynomial expression, we use the FOIL technique, (x  4)(x + 3). The expression can be rewritten as, x (x + 3)  4 (x + 3). On multiplying the outer terms we get, x^{2} + 3x  4x  12. This expression can be reduced to x^{2}  x  12.
Tips and Tricks
Important Notes
Example: Find the product of the polynomial expressions (2x+6) and (x8).
Solution: To simplify the product of polynomial expressions, we will use the FOIL technique.
Using the distributive property, the above polynomial expressions can be written as 2x(x 8) + 6(x  8).
Now applying the FOIL technique, we get, (2x^{2}  16x + 6x  48).
Now combining the like terms we get, 2x^{2}  10x  48. Hence, the product of polynomial expressions (2x+6) and (x8) on simplification gives: (2x^{2}  10x  48).
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