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Study Guides > Intermediate Algebra

Read: Define and Evaluate Polynomials

Learning Objectives

  • Identify the degree and leading coefficient of a polynomial
  • Evaluate a polynomial for given values
In the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a polynomial. A polynomial function is a function consisting of sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variables with an non-negative integer exponents. Non negative integers are 0,1,2,3,40, 1, 2, 3, 4, ... You may see a resemblance between expressions and polynomials, which we have been studying in this course.  Polynomials are a special sub-group of mathematical expressions and equations. The following table is intended to help you tell the difference between what is a polynomial and what is not.
IS a Polynomial Is NOT a Polynomial Because
2x212x92x^2-\frac{1}{2}x -9 2x2+x\frac{2}{x^{2}}+x Polynomials only have variables in the numerator
y4y3\frac{y}{4}-y^3 2y+4\frac{2}{y}+4 Polynomials only have variables in the numerator
12(a)+9\sqrt{12}\left(a\right)+9  a+7\sqrt{a}+7 Roots are equivalent to rational exponents, and polynomials only have integer exponents
The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient. The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent. A polynomial containing two terms, such as 2x92x - 9, is called a binomial. A polynomial containing three terms, such as 3x2+8x7-3{x}^{2}+8x - 7, is called a trinomial. We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents. 4x^3 - 9x^2 + 6x, with the text

Given a polynomial expression, identify the degree and leading coefficient.

  1. Find the highest power of x to determine the degree.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.

Example

For the following polynomials, identify the degree, the leading term, and the leading coefficient.
  1. 3+2x24x33+2{x}^{2}-4{x}^{3}
  2. 5t52t3+7t5{t}^{5}-2{t}^{3}+7t
  3. 6pp326p-{p}^{3}-2

Answer:

  1. The highest power of x is 33, so the degree is 33. The leading term is the term containing that degree, 4x3-4{x}^{3}. The leading coefficient is the coefficient of that term, 4-4.
  2. The highest power of t is 55, so the degree is 55. The leading term is the term containing that degree, 5t55{t}^{5}. The leading coefficient is the coefficient of that term, 55.
  3. The highest power of p is 33, so the degree is 33. The leading term is the term containing that degree, p3-{p}^{3}, The leading coefficient is the coefficient of that term, 1-1.

  In the following video example, we will identify the terms, leading coefficient, and degree of a polynomial. https://youtu.be/3u16B2PN9zk The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.
Monomials Binomials Trinomials Other Polynomials
1515 3y+133y+13 x3x2+1x^{3}-x^{2}+1 5x4+3x36x2+2x5x^{4}+3x^{3}-6x^{2}+2x
12x \displaystyle \frac{1}{2}x 4p74p-7 3x2+2x93x^{2}+2x-9 13x52x4+29x3x2+4x56\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}
4y3-4y^{3} 3x2+58x3x^{2}+\frac{5}{8}x 3y3+y223y^{3}+y^{2}-2 3t33t23t33t^{3}-3t^{2}-3t-3
16n416n^{4} 14y3+3y14y^{3}+3y a7+2a53a3a^{7}+2a^{5}-3a^{3} q7+2q53q3+qq^{7}+2q^{5}-3q^{3}+q
When the coefficient of a polynomial term is 00, you usually do not write the term at all (because 00 times anything is 00, and adding 00 doesn’t change the value). The last binomial above could be written as a trinomial, 14y3+0y2+3y14y^{3}+0y^{2}+3y. A term without a variable is called a constant term, and the degree of that term is 00. For example 1313 is the constant term in 3y+133y+13. You would usually say that 14y3+3y14y^{3}+3y has no constant term or that the constant term is 00.

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 3x22x+13x^{2}-2x+1 for x=1x=-1.

Answer: Substitute 1-1 for each x in the polynomial.

3(1)22(1)+13\left(-1\right)^{2}-2\left(-1\right)+1

Following the order of operations, evaluate exponents first.

3(1)2(1)+13\left(1\right)-2\left(-1\right)+1

Multiply 33 times 11, and then multiply 2-2 times 1-1.

3+(2)(1)+13+\left(-2\right)\left(-1\right)+1

Change the subtraction to addition of the opposite.

3+2+13+2+1

Find the sum.

Answer

3x22x+1=6[/latex],for[latex]x=13x^{2}-2x+1=6[/latex], for [latex]x=-1

Example

Evaluate 23p4+23p \displaystyle -\frac{2}{3}p^{4}+2^{3}-p for p=3p = 3.

Answer: Substitute 33 for each p in the polynomial.

23(3)4+2(3)33 \displaystyle -\frac{2}{3}\left(3\right)^{4}+2\left(3\right)^{3}-3

Following the order of operations, evaluate exponents first and then multiply.

23(81)+2(27)3 \displaystyle -\frac{2}{3}\left(81\right)+2\left(27\right)-3

Add and then subtract to get 3-3.

54+543-54 + 54 – 3

Answer

23p4+2p3p=3[/latex],for[latex]p=3 \displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=-3[/latex], for [latex]p = 3

 IN the following video we show more examples of evaluating polynomials for given values of the variable.

https://youtu.be/2EeFrgQP1hM

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