is y 3x 2 2x 2 x a polynomial function Polynomial or Not These are polynomials 3x x 2 6y 2 7 9 x 3xyz 3xy 2 z 0 1xz 200y 0 5 512v 5 99w 5 5 Yes 5 is a polynomial one term is allowed and it can be just a constant These are not polynomials 3xy 2 is not because the exponent is 2 exponents can only be 0 1 2
A polynomial is a special algebraic expression with terms that consist of real number coefficients and variable factors with whole number exponents color Cerulean Examples of polynomials 3x 2 quad 7xy 5 quad frac 3 2 x 3 3x 2 frac 1 2 x 1 quad 6x 2 y 4xy 3 4xy 3 7 We ve also seen that a binomial is the sum or difference of two monomial terms as in 3 x 5 x 2 4 or 3 x y 2 2 x 2 y We ve also seen that a trinomial is the sum or difference of three monomial terms as in x 2 2 x 3 or x 2 4 x y 5 y 2
is y 3x 2 2x 2 x a polynomial function
is y 3x 2 2x 2 x a polynomial function
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For example in f x 3 x 2 x 2 2 we can multiply 3 x from the first factor by x 2 in the second factor This gives a leading term of 3 x 3 The leading term of the polynomial is 3 x 3 and so the end behavior of function f will be the same as the end behavior of 3 x 3 A polynomial function in standard form is f x a n x n a n 1 x n 1 a 2 x 2 a 1 x a 0 This algebraic expression is called a polynomial function in variable x Here a n a n 1 a 0 are real number constants a n can t be equal to zero and is called the leading coefficient n is a non negative integer
Put more simply a function is a polynomial function if it is evaluated with addition subtraction multiplication and non negative integer exponents Polynomial functions can also be multivariable For example q x y 3x 2y 2xy 6x 9 is a polynomial function We can attempt to factor this polynomial to find solutions for f x 0 x 6 3x 4 2x 2 0 nonumber begin align x 2 x 4 3x 2 2 0 text Factoring out the greatest common factor x 2 x 2 1 x 2 2 0 text Factoring the inside as a quadratic in x 2 end align
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Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi valid function x y 1 valid function y 2 x 1 valid function x 1 Show More Description Check whether the input is a valid x 2 2x 1 cdot 2x 2 3x 6 long division frac x 3 x 2 x 2 x 2 factor 5a 2 30a 45 Show More Recognize characteristics of graphs of polynomial functions Use factoring to nd zeros of polynomial functions Identify zeros and their multiplicities Determine end behavior Understand the relationship between degree and turnin
A degree 2 polynomial in two variables is a function of the form p x y a2 0x2 a1 1xy a0 2y2 a1 0x a0 1y a0 0 where a2 0 a1 1 a0 2 a1 0 a0 1 a0 0 2 R as long as a2 0 a1 1 and a0 2 don t all equal 0 For example p x y 2x2 4xy 7y2 3x 2y 8 is a degree 2 polynomial in two variables Basic Algebra We may be able to solve using basic algebra Example 2x 1 2x 1 is a linear polynomial The graph of y 2x 1 is a straight line It is linear so there is one root Use Algebra to solve A root is when y is zero 2x 1 0 Subtract 1 from both sides 2x 1 Divide both sides by 2 x 1 2
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is y 3x 2 2x 2 x a polynomial function - For example in f x 3 x 2 x 2 2 we can multiply 3 x from the first factor by x 2 in the second factor This gives a leading term of 3 x 3 The leading term of the polynomial is 3 x 3 and so the end behavior of function f will be the same as the end behavior of 3 x 3