Cubic Polynomials: Definition, Formula, Method, Graphing & Examples

Cubic Polynomials: Polynomial is derived from the Greek word. “Poly” means many and “nomial” means terms, so together, we can call a polynomial as many terms. So, a polynomial has one or more than one number of terms but is not infinite. Polynomials are algebraic expressions that contain variables and constants with whole-number exponents. Cubic Polynomials, on the other hand, are polynomials of degree three.

A polynomial is classified into four forms based on its degree: zero polynomial, linear polynomial, quadratic polynomial, and cubic polynomial. A cubic polynomial has the generic form ax³ + bx² + cx + d, a ≠ 0. Where a, b, and c are coefficients and d is the constant, all of which are real integers. A cubic equation is an equation involving a cubic polynomial.

In this article, we will discuss the polynomials, their types, how to solve cubic polynomials, the graph of a cubic polynomial, and the relationship between the zeros and coefficients of a cubic polynomial. Continue reading to know more.

Polynomial Definition

Polynomials are algebraic expressions consisting of variables and constants with exponents of variables as whole numbers. Polynomials follow specific rules when it comes to mathematical operations. You can add, subtract, and multiply polynomials, and these operations will always give another polynomial as the result. A polynomial looks like this:

A polynomial \(p(x)\) in one variable \(x\) is an algebraic expression in \(x\) of the form:

\(p(x) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} +  \ldots  + {a_2}{x^2} + {a_1}x + {a_0}\)

Where \({a_0},\,{a_1},\,{a_2},\, \ldots ,\,{a_n}\) are constants, \(n \in W\) and \({a_n} \ne 0.\)

\({a_0},\,{a_1},\,{a_2},\, \ldots ,\,{a_n}\) are respectively the coefficients of \({x^0},{x^1},{x^2},…,{x^n}\) and \(n\) is called the degree of the polynomial. Each of \({a_n}{x^n},\,{a_{n – 1}}{x^{n – 1}},\, \ldots ,\,{a_0}\) with \({a_n} \ne 0\) is called a term of the polynomial \(p(x).\)

It is also important to note that a polynomial can’t have fractional or negative exponents. Examples of polynomials are \({3{y^2} + 2x + 5,\,{x^3} + 2{x^2} – 9x – 4,\,10{x^3} + 5x + y,\,4{x^2} – 5x + 7}\) etc.

Types of Polynomials

Based on the Number of Terms

(i) Monomial: A polynomial having only one term is called a monomial. 
Examples of monomials are; \(5,\,2x,\,3{a^2},\,4xy\), etc.

(ii) Binomial: A polynomial having two terms separated by either the addition \((+)\) or subtraction sign \((-)\) is called a binomial. 
Examples of binomial expressions are \(2x + 3,\,3x – 1,\,2x + 5y,\,6x – 3y,\) etc.

(iii) Trinomial: A polynomial having exactly three terms is called trinomial. 
Examples of trinomials are: \(4{x^2} + 9x + 7,\,12pq + 4{x^2} – 10,\,3x + 5{x^2} – 6{x^3}\) etc.

Based on the Degree of a Polynomial

(i) Constant or Zero Polynomial: A polynomial whose power of the variable is zero is known as a constant or zero polynomial. When the power of the variable is zero, its value is nothing but \(1\) as \({x^0} = 1\). The zero polynomials will have terms that are constants like \(2, 5, 10, 101,\) etc.
Example: \(3{x^0} = 3 \times 1 = 3\)

(ii) Linear Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(1\) is a linear polynomial.
Example: \(x – 1,\,y + 1,\,a + 4,\) etc.

(iii) Quadratic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(2\) is a quadratic polynomial.
Example: \({x^2} + x,\,{y^2} + 1,\,{a^2} + 8,\) etc.

(iv) Cubic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(3\) is a cubic polynomial.
Example: \({y^3} + 8,\,{x^3} – 27,\,5 + {a^3},\,{x^3} + {x^2} – x + 2\) etc.

(v) Quartic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(4\) is known as a quartic polynomial or fourth-degree polynomial.
Example: \({x^4} + {x^3} – {x^2} + x + 1,\,{y^4} – {y^2} + 1,\) etc.

Zeros of a Polynomial

Let \(p(x)\) be a polynomial. If \(p(a)=0\), then we say that \(a\) is a zero of the polynomial \(p(x).\)

Remainder Theorem

Let \(p(x)\) be a polynomial of degree \(1\) or more and let \(a\) be any real number. If \(p(x)\) is divided by \((x-a)\) then the remainder is \(p(a).\)

Factors of a Polynomial

Let \(p(x)\) be a polynomial of degree \(1\) or more and let \(a\) be any real number.
(i) If \(p(a)=0\), then \((x-a)\) is a factor of \(p(x).\)
(ii) If \((x-a)\) is a factor of \(p(x)\) then \(p(a)=0.\)

Cubic Polynomial in One Variable

A cubic polynomial in one variable is a polynomial of the form \(a{x^3} + b{x^2} + cx + d,\) where the coefficients \(a, b, c,\) and \(d\) are real numbers, and the variable \(x\) takes real values.
A cubic polynomial is a polynomial of degree \(3.\) 
An equation involving a cubic polynomial is called a cubic equation.

Solving a Cubic Polynomial

A cubic polynomial has three roots which can be found by using the trial and error method followed by the long division method or by factorisation method.
Here we will learn using an example how to solve a cubic polynomial.
Example: Find the roots of the polynomial \(2{x^3} + 3{x^2} – 11x – 6.\)

Step 1: First, use the factor theorem to check the possible values by the trial-and-error method.
Let \(f(x) = 2{x^3} + 3{x^2} – 11x – 6\)
\(f\left( 1 \right) = 2 + 3 – 11 – 6 \ne 0\)
\(f\left( { – 1} \right) =  – 2 + 3 + 11 – 6 \ne 0\)
\(f\left( 2 \right) = 16 + 12 – 22 – 6 = 0\)
We find that the root is \(2\). Hence, \(x-2\) is the factor of \(2 x^{3}+3 x^{2}-11 x-6\)

Step 2: Find the other roots either by inspection or by the long division method.
\(2{x^3} + 3{x^2} – 11x – 6 = (x – 2)\left( {2{x^2} + 7x + 3} \right) = (x – 2)(2x + 1)(x + 3)\)
So, the roots are \(x = 2,\, – \frac{1}{2},\, – 3\)

Graph of Cubic Polynomials

Here we will discuss the graph of a cubic polynomial in one variable, which are of the form
\(a x^{3}+b x^{2}+c x+d\)
Where \(a, b, c\) and \(d\) are real numbers and \(a≠0.\)
If the sign of \(a\) is positive, then the graph comes from down and goes up.
If the sign of \(a\) is negative, then the graph comes from up and goes down.

For example: Draw the graph of the cubic polynomial \(-x^{3}+6 x^{2}-12 x+8\)
Let \(f(x)=-x^{3}+6 x^{2}-12 x+8=-(x-2)^{3}\)
The \(y\)- intercept is given by \((0, f(0))=(0,8)\)
The zeros of \(f\) are solutions of the equation \((x-2)^{3}=0\)

Function \(f\) has one zero at \(x = 2\) of multiplicity three, and therefore the graph of \(f\) cuts the \(x\) axis at \(x=2.\)
After the expansion of \(f(x),\) we can see that the coefficient (of \({x^3}\)) is negative; the graph of \(f\) goes downward direction on the right-hand and upward direction on the left-hand side.
At \(x=2,\) the graph cuts the \(x\)-axis. 

The \(y\)-intercept is a point on the graph of \(f\). 
Also, the graph of \(f(x)=-(x-2)^{3}\) is that of \(f\left( x \right) = {x^3}\) shifted \(2\) units to the right because of the term \((x-2)\) and reflected on the \(x\)-axis because of the negative sign in \(f\left( x \right) = \, – {\left( {x – 2} \right)^3}.\)
Hence, the graph of \( – {x^3} + 6{x^2} – 12x + 8\) will be as shown below

Relationship between Zeroes and Coefficients of a Cubic Polynomial

If \(α, β\) and \(\gamma \) are the roots of a cubic polynomial \(a{x^3} + b{x^2} + cx + d\), then

\(\alpha + \beta + \gamma = – \frac{b}{a}\)

\(\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}\)

\(\alpha \beta \gamma = – \frac{d}{a}\)

Solved Examples – Cubic Polynomials

Q.1. Find a cubic polynomial with the sum of zeroes, the sum of the product of its zeros taken two at a time, and the product of its zeros as \(2, -7, -14,\) respectively.
Ans: Let the polynomial be \(a{x^3} + b{x^2} + cx + d\) and the zeroes be \(α, β\) and \(\gamma \)
Then, \(\alpha + \beta + \gamma = \, – \frac{{ – 2}}{1} = 2 = \, – \frac{b}{a}\)
\(\alpha \beta + \beta \gamma + \gamma \alpha = \, – 7 = \, – \frac{7}{1} = \frac{c}{a}\)
\(\alpha \beta \gamma = \, – 14 = \, – \frac{{14}}{1} = \, – \frac{d}{a}\)
\(\therefore {\mkern 1mu} \,a = 1,{\mkern 1mu} \,b = \, – 2,{\mkern 1mu} \,c = \, – 7\) and \(d=14\)
So, one cubic polynomial which satisfies the given conditions will be \({x^3} – 2{x^2} – 7x + 14.\)

Q.2. Solve the cubic equation \(x^{3}-7 x^{2}+4 x+12=0\)
Ans: Let \(f(x) = {x^3} – 7{x^2} + 4x + 12\)
The possible values are \(\pm 1,\, \pm 2,\, \pm 3,\, \pm 4,\, \pm 5,\, \pm 6,\, \pm 12\)
We find that \({\rm{f}}\left( { – 1} \right) = \, – 1 – 7 – 4 + 12 = 0\)
So, \((x + 1)\) is a factor of \(f(x.\)) 
Using long division method we get
\({x^3} – 7{x^2} + 4x + 12\)
\(= (x + 1)\left({{x^2} – 8x + 12} \right)\)
\(=(x+1)(x-2)(x-6)\)
So, the roots are \(–1, 2, 6.\)

Q.3. Solve \({x^3} – 6{x^2} + 11x – 6 = 0\)
Ans: We can factorise this equation to give
\((x-1)(x-2)(x-3)=0\)
This equation has three real roots, all different \(–\) the solutions are \(x=1, x=2\) and \(x=3.\)

Q.4. Draw the graph of \(y=x^{3}+3\) for \(-3 \leq x \leq 3\). Use your graph to find the value of \(y\) when \(x=2.5.\)
Ans: Take the value of \(x\) from \(-3\) to \(3\) and put in the given equation \(y=x^{3}+3\) to get the values of \(y.\)

When \(x=2.5, y≈18.6\)

Q.5. Show that \((2x-3)\) is a factor of \(\left(x+2 x^{3}-9 x^{2}+12\right)\)
Ans: Let \(p(x)=2 x^{3}-9 x^{2}+x+12\) and \(g(x)=2 x-3\)
Now, \(g(x)=0 \Rightarrow 2 x-3=0 \Rightarrow 2 x=3 \Rightarrow x=\frac{3}{2}\)
By factor theorem, \(g(x)\) will be a factor of \(p(x)\), if \(p\left(\frac{3}{2}\right)=0\)
Now, \(p\left( {\frac{3}{2}} \right) = \left\{ {2 \times {{\left( {\frac{3}{2}} \right)}^3} – 9 \times {{\left( {\frac{3}{2}} \right)}^2} + \frac{3}{2} + 12} \right\}\)
\( = \left\{ {\left( {2 \times \frac{{27}}{8}} \right) – \left( {9 \times \frac{9}{4}} \right) + \frac{3}{2} + 12} \right\}\)
\(=\left(\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12\right)=\left(\frac{27-81+6+48}{4}\right)=\frac{0}{4}=0\)
Since \(p\left(\frac{3}{2}\right)=0\), so \(g(x)\) is a factor of \(p(x).\)

Summary

In this article, we have learned about polynomials, how it looks like, types of polynomials based on the number of terms like monomials, binomials, and trinomials. And based on the degree, polynomials are further classified into zero-degree polynomial or constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial, quartic polynomial, etc. Then we have discussed in detail the cubic polynomials, their graph, zeros, and their factors, and solved examples.

Frequently Asked Questions

We have provided some frequently asked questions on Cubic Polynomials here:

We hope that our article on cubic polynomials was useful for you. If you have any query or feedback to share with us, please feel to drop a comment below. We will get back to you at the earliest. In the end, we are winding up this article with best wishes for your exams on behalf of Embibe.