5.2: Introduction to Polynomials (2024)

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    • 5.2: Introduction to Polynomials (1)
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    Learning Objectives

    • Identify a polynomial and determine its degree.
    • Evaluate a polynomial for given values of the variables.
    • Evaluate a polynomial using function notation.

    Definitions

    A polynomial is a special algebraic expression with terms that consist of real number coefficients and variable factors with whole number exponents.

    \(\color{Cerulean}{Examples\:of\:polynomials:}\)

    \(3x^{2}\quad 7xy+5\quad \frac{3}{2}x^{3}+3x^{2}-\frac{1}{2}x+1\quad 6x^{2}y-4xy^{3}-4xy^{3}+7\)

    Polynomials do not have variables in the denominator of any term.

    \(\color{Cerulean}{Examples\:that\:are\:not\:polynomials:}\)

    \(\frac{2x^{2}}{y} \quad 5\sqrt{x}+5\quad 5x^{2}+3x^{-2}+7\quad \frac{2}{x}-\frac{5}{y}=3\)

    The degree of a term in a polynomial is defined to be the exponent of the variable, or if there is more than one variable in the term, the degree is the sum of their exponents. Recall that \(x^{0}=1\); any constant term can be written as a product of \(x^{0}\) and itself. Hence the degree of a constant term is \(0\).

    Term Degree
    \(3x^{2}\) \(2\)
    \(6x^{2}y\) \(2+1=3\)
    \(7a^{2}b^{3}\) \(2+3=5\)
    \(8\) \(0\), since \(8=8x^{0}\)
    \(2x\) \(1\), since \(x=x^{1}\)
    Table \(\PageIndex{1}\)

    The degree of a polynomial is the largest degree of all of its terms.

    Polynomial Degree
    \(4x^{5}-3x^{3}+2x-1\) \(5\)
    \(6x^{2}y-5xy^{3}+7\) \(4\), because \(5xy^{3}\) has degree \(4\).
    \(12x+54\) \(1\), because \(x=x^{1}\)
    Table \(\PageIndex{2}\)

    We classify polynomials by the number of terms and the degree as follows:

    Expression Classification Degree
    \(5x^{7}\) Monomial (one term) \(7\)
    \(8x^{6}-1\) Binomial (two terms) \(6\)
    \(-3x^{2}+x-1\) Trinomial (three terms) \(2\)
    \(5x^{3}-2x^{2}+3x-6\) Polynomial (many terms) \(3\)
    Table \(\PageIndex{3}\)

    In this text, we will call polynomials with four or more terms simply polynomials.

    Example \(\PageIndex{1}\)

    Classify and state the degree:

    \(7x^{2}−4x^{5}−1\).

    Solution:

    Here there are three terms. The highest variable exponent is \(5\). Therefore, this is a trinomial of degree \(5\).

    Answer:

    Trinomial; degree \(5\)

    Example \(\PageIndex{2}\)

    Classify and state the degree:

    \(12a^{5}bc^{3}\).

    Solution:

    Since the expression consists of only multiplication, it is one term, a monomial. The variable part can be written as \(a^{5}b^{1}c^{3}\); hence its degree is \(5+1+3=9\).

    Answer:

    Monomial; degree \(9\)

    Example \(\PageIndex{3}\)

    Classify and state the degree:

    \(4x^{2}y−6xy^{4}+5x^{3}y^{3}+4\).

    Solution:

    The term \(4x^{2}y\) has degree \(3\); \(−6xy^{4}\) has degree \(5; 5x^{3}y^{3}\) has degree \(6\); and the constant term \(4\) has degree \(0\). Therefore, the polynomial has \(4\) terms with degree \(6\).

    Answer:

    Polynomial; degree \(6\)

    Of particular interest are polynomials with one variable, where each term is of the form \(a_{n}x^{n}\). Here \(a_{n}\) is any real number and \(n\) is any whole number. Such polynomials have the standard form

    \[a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\]

    Typically, we arrange terms of polynomials in descending order based on the degree of each term. The leading coefficient is the coefficient of the variable with the highest power, in this case, \(a_{n}\).

    Example \(\PageIndex{4}\)

    Write in standard form:

    \(3x−4x^{2}+5x^{3}+7−2x^{4}\).

    Solution:

    Since terms are separated by addition, write the following:

    \(\begin{aligned} & 3x-4x^{2}+5x^{3}+7-2x^{4} \\ &=3x+(-4x^{2})+5x^{3}+7+(-2x^{4}) \end{aligned}\)

    In this form, we can see that the subtraction in the original corresponds to negative coefficients. Because addition is commutative, we can write the terms in descending order based on the degree of each term as follows:

    \(\begin{aligned} &=(-2x^{4})+5x^{3}+(-4x^{2})+3x+7 \\ &=-2x^{4}+5x^{3}-4x^{2}+3x+7 \end{aligned}\)

    Answer:

    \(-2x^{4}+5x^{3}-4x^{2}+3x+7\)

    We can further classify polynomials with one variable by their degree as follows:

    Polynomial Name
    \(5\) Constant (degree \(0\) )
    \(2x+1\) Linear (degree \(1\) )
    \(3x^{2}+5x-3\) Quadratic (degree \(2\) )
    \(x^{3}+x^{2}+x+1\) Cubic (degree \(3\) )
    \(7x^{4}+3x^{3}-7x+8\) Fourth-degree polynomial
    Table \(\PageIndex{4}\)

    In this text, we call any polynomial of degree \(n≥4\) an \(n\)th-degree polynomial. In other words, if the degree is \(4\), we call the polynomial a fourth-degree polynomial. If the degree is \(5\), we call it a fifth-degree polynomial, and so on.

    Evaluating Polynomials

    Given the values for the variables in a polynomial, we can substitute and simplify using the order of operations.

    Example \(\PageIndex{5}\)

    Evaluate:

    \(3x−1\), where \(x=−\frac{3}{2}\).

    Solution:

    First, replace the variable with parentheses and then substitute the given value.

    Answer:

    \(-\frac{11}{2}\)

    Example \(\PageIndex{6}\)

    Evaluate:

    \(3x^{2}+2x−1\), where \(x=−1\).

    Solution:

    Answer:

    \(0\)

    Example \(\PageIndex{7}\)

    Evaluate:

    \(−2a^{2}b+ab^{2}−7\), where \(a=3\) and \(b=−2\).

    Solution:

    Answer:

    \(41\)

    Example \(\PageIndex{8}\)

    The volume of a sphere in cubic units is given by the formula \(V=\frac{4}{3}πr^{3}\), where \(r\) is the radius. Calculate the volume of a sphere with radius \(r=\frac{3}{2}\) meters.

    5.2: Introduction to Polynomials (2)

    Solution:

    \(\begin{aligned} V&=\frac{4}{3}\pi r^{3} \\ &=\frac{4}{3}\pi \left( \frac{3}{2} \right)^{3} \\ &=\frac{4}{3}\pi \frac{3^{3}}{2^{3}} \\ &=\frac{\color{Cerulean}{\stackrel{1}{\cancel{\color{black}{4}}}}}{\color{Cerulean}{\stackrel{\cancel{\color{black}{3}}}{1}}} \pi \frac{\color{Cerulean}{\stackrel{9}{\cancel{\color{black}{27}}}}}{\color{Cerulean}{\stackrel{\cancel{\color{black}{8}}}{2}}} \\ &=\frac{9}{2} \pi \end{aligned}\)

    Answer:

    \(\frac{9}{2}\pi\) cubic meters

    Exercise \(\PageIndex{1}\)

    Evaluate:

    \(x^{3}−x^{2}+4x−2\), where \(x=−3\).

    Answer

    \(-50\)

    Polynomial Functions

    Polynomial functions with one variable are functions that can be written in the form

    \[f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{0}\],

    where \(a_{n}\) is any real number and \(n\) is any whole number. Some examples of the different classes of polynomial functions are listed below:

    Polynomial function Name
    \(f(x)=5\) Constant function (degree \(0\) )
    \(f(x)=-2x+1\) Linear function (degree \(1\) )
    \(f(x)=5x^{2}+4x-3\) Quadratic function (degree \(2\) )
    \(f(x)=x^{3}-1\) Cubic function (degree \(3\) )
    \(f(x)=4x^{5}+3x^{4}-7\) Polynomial function
    Table \(\PageIndex{5}\)

    Since there are no restrictions on the values for \(x\), the domain of any polynomial function consists of all real numbers.

    Example \(\PageIndex{9}\)

    Calculate:

    \(f(5)\), given \(f(x)=−2x^{2}+5x+10\).

    Solution:

    Recall that the function notation \(f(5)\) indicates we should evaluate the function when \(x=5\). Replace every instance of the variable \(x\) with the value \(5\).

    Answer:

    \(f(5)=-15\)

    Example \(\PageIndex{10}\)

    Calculate:

    \(f(−1)\), given \(f(x)=−x^{3}+2x^{2}−4x+1\).

    Solution:

    Replace the variable \(x\) with \(−1\).

    \(\begin{aligned} f(\color{OliveGreen}{-1}\color{black}{)} &=-(\color{OliveGreen}{-1}\color{black}{)^{3}+2(}\color{OliveGreen}{-1}\color{black}{)^{2}-4(}\color{OliveGreen}{-1}\color{black}{)+1} \\ &=-(-1)+2\cdot 1 +4+1 \\ &=1+2+4+1 \\ &=8 \end{aligned}\)

    Answer:

    \(f(-1)=8\)

    Exercise \(\PageIndex{2}\)

    Given \(g(x)=x^{3}−2x^{2}−x−4\), calculate \(g(−1)\).

    Answer

    \(g(−1)=−6\)

    Key Takeaways

    • Polynomials are special algebraic expressions where the terms are the products of real numbers and variables with whole number exponents.
    • The degree of a polynomial with one variable is the largest exponent of the variable found in any term.
    • The terms of a polynomial are typically arranged in descending order based on the degree of each term.
    • When evaluating a polynomial, it is a good practice to replace all variables with parentheses and then substitute the appropriate values.
    • All polynomials are functions.

    Exercise \(\PageIndex{3}\) Definitions

    Classify the given polynomial as linear, quadratic, or cubic.

    1. \(2x+1\)
    2. \(x^{2}+7x+2\)
    3. \(2−3x^{2}+x\)
    4. \(4x\)
    5. \(x^{2}−x^{3}+x+1\)
    6. \(5−10x^{3}\)
    Answer

    1. Linear

    3. Quadratic

    5. Cubic

    Exercise \(\PageIndex{4}\) Definitions

    Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

    1. \(x^{3}−1\)
    2. \(x^{2}y^{2}\)
    3. \(x−x^{5}+1\)
    4. \(x^{2}+3x−1\)
    5. \(5ab^{4}\)
    6. \(13x−12\)
    7. \(−5x^{3}+2x+1\)
    8. \(8x^{2}−9\)
    9. \(4x^{5}−5x^{3}+6x\)
    10. \(8x^{4}−x^{5}+2x−3\)
    11. \(9x+7\)
    12. \(x^{5}+x^{4}+x^{3}+x^{2}−x+1\)
    13. \(6x−1+5x^{4}−8\)
    14. \(4x−3x^{2}+3\)
    15. \(7\)
    16. \(x^{2}\)
    17. \(4x^{2}y−3x^{3}y^{3}+xy^{3}\)
    18. \(a^{3}b^{2}−6ab\)
    19. \(a^{3}b^{3}\)
    20. \(x^{2}y−y^{2}x\)
    21. \(xy−3\)
    22. \(a^{5}bc^{2}+3a^{9}−5a^{4}b^{3}c\)
    23. \(−3x^{10}y^{2}z−xy^{12}z+9x^{13}+30\)
    24. \(7x^{0}\)
    Answer

    1. Binomial; degree \(3\)

    3. Trinomial; degree \(5\)

    5. Monomial; degree \(5\)

    7. Trinomial; degree \(3\)

    9. Trinomial; degree \(5\)

    11. Binomial; degree \(1\)

    13. Not a polynomial

    15. Monomial; degree \(0\)

    17. Trinomial; degree \(6\)

    19. Monomial; degree \(6\)

    21. Binomial; degree \(2\)

    23. Polynomial; degree \(14\)

    Exercise \(\PageIndex{5}\) Definitions

    Write the following polynomials in standard form.

    1. \(1−6x+7x^{2}\)
    2. \(x−9x^{2}−8\)
    3. \(7−x^{3}+x^{7}−x^{2}+x−5x^{5}\)
    4. \(a^{3}−a^{9}+6a^{5}−a+3−a^{4}\)
    Answer

    1. \(7x^{2}−6x+1\)

    3. \(x^{7}−5x^{5}−x^{3}−x^{2}+x+7\)

    Exercise \(\PageIndex{6}\) Evaluating Polynomials

    1. Fill in the following chart:
      5.2: Introduction to Polynomials (3)
      Figure \(\PageIndex{2}\)
    2. Fill in the following chart:
      5.2: Introduction to Polynomials (4)
      Figure \(\PageIndex{3}\)
    Answer

    1.

    5.2: Introduction to Polynomials (5)

    Exercise \(\PageIndex{7}\) Evaluating Polynomials

    Evaluate.

    1. \(2x−3\), where \(x=3\)
    2. \(x^{2}−3x+5\), where \(x=−2\)
    3. \(−12x+13\), where \(x=−13\)
    4. \(−x^{2}+5x−1\), where \(x=−12\)
    5. \(−2x^{2}+3x−5\), where \(x=0\)
    6. \(8x^{5}−27x^{3}+81x−17\), where \(x=0\)
    7. \(y^{3}−2y+1\), where \(y=−2\)
    8. \(y^{4}+2y^{2}−32\), where \(y=2\)
    9. \(a^{3}+2a^{2}+a−3\), where \(a=−3\)
    10. \(x^{3}−x^{2}\), where \(x=5\)
    11. \(34x^{2}−12x+36\), where \(x=−23\)
    12. \(58x^{2}−14x+12\), where \(x=4\)
    13. \(x^{2}y+xy^{2}\), where \(x=2\) and \(y=−3\)
    14. \(2a^{5}b−ab^{4}+a^{2}b^{2}\), where \(a=−1\) and \(b=−2\)
    15. \(a^{2}−b^{2}\), where \(a=5\) and \(b=−6\)
    16. \(a^{2}−b^{2}\), where \(a=34\) and \(b=−14\)
    17. \(a^{3}−b^{3}\), where \(a=−2\) and \(b=3\)
    18. \(a^{3}+b^{3}\), where \(a=5\) and \(b=−5\)
    Answer

    1. \(3\)

    3. \(\frac{1}{2}\)

    5. \(−5\)

    7. \(−3\)

    9. \(−15\)

    11. \(\frac{7}{6}\)

    13. \(6\)

    15. \(−11\)

    17. \(−35\)

    Exercise \(\PageIndex{8}\) Evaluating Polynomials

    For each problem, evaluate \(b^{2}−4ac\), given the following values.

    1. \(a=−1, b=2\), and \(c=−1\)
    2. \(a=2, b=−2\), and \(c=12\)
    3. \(a=3, b=−5, c=0\)
    4. \(a=1, b=0\), and \(c=−4\)
    5. \(a=14, b=−4\), and \(c=2\)
    6. \(a=1, b=5\), and \(c=6\)
    Answer

    1. \(0\)

    3. \(25\)

    5. \(14\)

    Exercise \(\PageIndex{9}\) Evaluating Polynomials

    The volume of a sphere in cubic units is given by the formula \(V=\frac{4}{3}πr^{3}\), where \(r\) is the radius. For each problem, calculate the volume of a sphere given the following radii.

    1. \(r = 3\) centimeters
    2. \(r = 1\) centimeter
    3. \(r = \frac{1}{2}\) feet
    4. \(r = \frac{3}{2}\) feet
    5. \(r = 0.15\) in
    6. \(r = 1.3\) inches
    Answer

    1. \(36π\) cubic centimeters

    3. \(\frac{π}{6}\) cubic feet

    5. \(0.014\) cubic inches

    Exercise \(\PageIndex{10}\) Evaluating Polynomials

    The height in feet of a projectile launched vertically from the ground with an initial velocity \(v_{0}\) in feet per second is given by the formula \(h=−16t^{2}+v_{0}t\), where \(t\) represents time in seconds. For each problem, calculate the height of the projectile given the following initial velocity and times.

    1. \(v_{0}=64\) feet/second, at times \(t = 0, 1, 2, 3, 4\) seconds
    2. \(v_{0}=80\) feet/second, at times \(t = 0, 1, 2, 2.5, 3, 4, 5\) seconds
    Answer

    1.

    Time Height
    \(t=0\) seconds \(h=0\) feet
    \(t=1\) second \(h=48\) feet
    \(t=2\) seconds \(h=64\) feet
    \(t=3\) seconds \(h=48\) feet
    \(t=4\) seconds \(h=0\) feet
    Table \(\PageIndex{6}\)

    Exercise \(\PageIndex{11}\) Evaluating Polynomials

    The stopping distance of a car, taking into account an average reaction time, can be estimated with the formula \(d=0.05v^{2}+1.5\), where \(d\) is in feet and \(v\) is the speed in miles per hour. For each problem, calculate the stopping distance of a car traveling at the given speeds.

    1. \(20\) miles per hour
    2. \(40\) miles per hour
    3. \(80\) miles per hour
    4. \(100\) miles per hour
    Answer

    1. \(21.5\) feet

    3. \(321.5\) feet

    Exercise \(\PageIndex{12}\) Polynomial Functions

    Given the linear function \(f(x)=\frac{2}{3}x+6\), evaluate each of the following.

    1. \(f(−6)\)
    2. \(f(−3)\)
    3. \(f(0)\)
    4. \(f(3)\)
    5. Find \(x\) when \(f(x)=10\).
    6. Find \(x\) when \(f(x)=−4\).
    Answer

    1. \(2\)

    3. \(6\)

    5. \(x=6\)

    Exercise \(\PageIndex{13}\) Polynomial Functions

    Given the quadratic function \(f(x)=2x^{2}−3x+5\), evaluate each of the following.

    1. \(f(−2)\)
    2. \(f(−1)\)
    3. \(f(0)\)
    4. \(f(2)\)
    Answer

    1. \(19\)

    3. \(5\)

    Exercise \(\PageIndex{14}\) Polynomial Functions

    Given the cubic function \(g(x)=x^{3}−x^{2}+x−1\), evaluate each of the following.

    1. \(g(−2)\)
    2. \(g(−1)\)
    3. \(g(0)\)
    4. \(g(1)\)
    Answer

    1. \(-15\)

    3. \(-1\)

    Exercise \(\PageIndex{15}\) Polynomial Functions

    The height in feet of a projectile launched vertically from the ground with an initial velocity of \(128\) feet per second is given by the function \(h(t)=−16t^{2}+128t\), where \(t\) is in seconds. Calculate and interpret the following.

    1. \(h(0)\)
    2. \(h(12) \)
    3. \(h(1) \)
    4. \(h(3) \)
    5. \(h(4) \)
    6. \(h(5) \)
    7. \(h(7) \)
    8. \(h(8)\)
    Answer

    1. The projectile is launched from the ground.

    3. The projectile is \(112\) feet above the ground \(1\) second after launch.

    5. The projectile is \(256\) feet above the ground \(4\) seconds after launch.

    7. The projectile is \(112\) feet above the ground \(7\) seconds after launch.

    Exercise \(\PageIndex{16}\) Discussion Board Topics

    1. Find and share some graphs of polynomial functions.
    2. Explain how to convert feet per second into miles per hour.
    3. Find and share the names of fourth-degree, fifth-degree, and higher polynomials.
    Answer

    1. Answers may vary

    3. Answers may vary

    5.2: Introduction to Polynomials (2024)

    FAQs

    5.2: Introduction to Polynomials? ›

    Polynomials are special algebraic expressions where the terms are the products of real numbers and variables with whole number exponents. The degree of a polynomial with one variable is the largest exponent of the variable found in any term.

    What is the basic introduction of polynomials? ›

    In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

    What are 5 examples of polynomials? ›

    Polynomial
    • 2a + 5b is a polynomial of two terms in two variables a and b.
    • 3xy + 5x + 1 is a polynomial of three terms in two variables x and y.
    • 3y4 + 2y3 + 7y2 – 9y + 3/5 is a polynomial of five terms in two variables x and y.
    • m + 5mn – 7m2n + nm2 + 9 is a polynomial of four terms in two variables m and n.

    What is 5 in polynomials? ›

    A fifth-degree polynomial is a polynomial in which the highest exponent of the variable is 5. This is a mathematical expression that has a variable raised to the fifth power as its highest term. It is also known as a quintic function. When graphed, these functions look similar to cubic functions.

    How to solve polynomials? ›

    To solve a polynomial equation, first write it in standard form. Once it is equal to zero, factor it and then set each variable factor equal to zero. The solutions to the resulting equations are the solutions to the original. Not all polynomial equations can be solved by factoring.

    What is a polynomial in math? ›

    Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.

    What is the formula of a polynomial? ›

    Constant Polynomial Function: P(x) = a = ax. Zero Polynomial Function: P(x) = 0; where all ai's are zero, i = 0, 1, 2, 3, …, n. Linear Polynomial Function: P(x) = ax + b. Quadratic Polynomial Function: P(x) = ax2+bx+c.

    How to identify a polynomial? ›

    The polynomials can be identified by noting which expressions contain only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The non-polynomial expressions will be the expressions which contain other operations. Explain why the non-polynomial expressions are not polynomials.

    What Cannot be a polynomial? ›

    While a polynomial can appear in many different ways, there are some rules about what is not considered a polynomial. A polynomial is NOT: An equation which contains division by a variable. An equation that contains negative exponents. An equation that contains fractional exponents.

    How to simplify polynomials? ›

    Polynomials can be simplified by using the distributive property to distribute the term on the outside of the parentheses by multiplying it by everything inside the parentheses. You can simplify polynomials by using FOIL to multiply binomials times binomials.

    How to multiply polynomials? ›

    Multiplying Polynomials
    1. First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
    2. Add the powers of the same variables using the exponent rule.
    3. Then, simplify the resulting polynomial by adding or subtracting the like terms.

    What is the basic form of a polynomial? ›

    A polynomial is a function of the form f(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0 . The degree of a polynomial is the highest power of x in its expression. Constant (non-zero) polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1, 2 , 3 and 4 respectively.

    What is the key concept of polynomial? ›

    Key Concepts

    A polynomial is a sum of terms each consisting of a variable raised to a nonnegative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term.

    What is the basic equation of a polynomial? ›

    Constant Polynomial Function: P(x) = a = ax. Zero Polynomial Function: P(x) = 0; where all ai's are zero, i = 0, 1, 2, 3, …, n. Linear Polynomial Function: P(x) = ax + b. Quadratic Polynomial Function: P(x) = ax2+bx+c.

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    Introduction: My name is Kieth Sipes, I am a zany, rich, courageous, powerful, faithful, jolly, excited person who loves writing and wants to share my knowledge and understanding with you.