Define and Evaluate Polynomials
Learning Outcomes
- Define, evaluate, and simplify single variable polynomials
IS a Polynomial | Is NOT a Polynomial | Reason |
Polynomials only have variables in the numerator | ||
Polynomials only have variables in the numerator | ||
Roots are equivalent to rational exponents, and polynomials only have integer exponents on variables |

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How to: Given a polynomial expression, identify the degree and leading coefficient
- Find the highest power of x to determine the degree.
- Identify the term containing the highest power of x to find the leading term.
- Identify the coefficient of the leading term.
Example
For the following polynomials, identify the degree, the leading term, and the leading coefficient.Answer:
- The highest power of x is , so the degree is . The leading term is the term containing that degree, . The leading coefficient is the coefficient of that term, .
- The highest power of t is , so the degree is . The leading term is the term containing that degree, . The leading coefficient is the coefficient of that term, .
- The highest power of p is , so the degree is . The leading term is the term containing that degree, , The leading coefficient is the coefficient of that term, .
Monomials | Binomials | Trinomials | Other Polynomials |
Evaluate a Polynomial
You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.Example
Evaluate for .Answer: Substitute for each x in the polynomial.
Following the order of operations, evaluate exponents first.
Multiply times , and then multiply times .
Find the sum:
Example
Evaluate for .Answer: Substitute for each p in the polynomial.
Following the order of operations, evaluate exponents first and then multiply.
Add and then subtract to get .
In the following video, we show more examples of evaluating polynomials for given values of the variable.
https://youtu.be/2EeFrgQP1hM