A zero polynomial is a special type of polynomial where every term has a coefficient of zero. That means the entire expression equals 0, no matter what value you put for the variable. Since 0 is a constant number, it can also be called a constant polynomial.

A zero-degree polynomial is a polynomial where the variable (like x or y) is raised to the power of 0. This means it behaves like a number (a constant), not a variable.

Key difference:

• A zero polynomial has all coefficients equal to 0.
• A zero-degree polynomial has no variable part, only a constant number (not zero).

What is a Zero Polynomial?

A zero polynomial is a polynomial whose value is zero. It is a constant polynomial with a constant function of value 0 and is expressed as P(x)=0. Since a zero polynomial has no terms, therefore the degree of a zero polynomial is undefined. However, many mathematicians define the degree of a zero polynomial to be negative, which is usually taken as -1 or –.

A function that defines a zero polynomial is called a constant function. It is also known as a zero map. The constant function or the function of a zero polynomial is expressed as P(x)=0 where x is the polynomial variable whose coefficient is 0 for each term.

A zero polynomial can have an infinite number of terms with the variables having different powers, only if all those variables have a coefficient of 0. 

Examples of a zero polynomial: 0, 0x+0, \(\Rightarrow 0x^2+0x+0\)

Degree of Zero Polynomial

Degree of a polynomial is the value of the highest power of the non-zero term of the polynomial. Degree of a zero polynomial is usually undefined unless it is assigned then it becomes -1 or (\ \infty \). The degree of a polynomial is the power of the non-zero term, but we do not have any non-zero term in a zero polynomial. Therefore, we do not have any term with us to calculate the degree of a polynomial.

A zero polynomial can be written as: (\f(x) = ax^0\) where (\a \ne 0\).

Zero of Zero Polynomial

Zero of any polynomial is the number that when substituted with the variable gives the value of the polynomial as zero. This number can be a rational number, an irrational number or a complex number.

However, in a zero polynomial coefficient of every term is zero, so even after substituting the values the polynomial will always be zero. So, zero is itself the zero polynomial.

What is a Zero Degree Polynomial?

Zero degree polynomial is any polynomial in which the degrees of all the variables are equal to zero. Therefore, a zero-degree polynomial can only consist of a single constant term.

Examples of a zero degree polynomial

\(\Rightarrow 4x^0-2x^0=2\)

\(\Rightarrow 5x^0=5\)

\(\Rightarrow -9x^0= -9\)

Therefore, a zero degree polynomial can be any number positive or negative but without a variable.

Learn about Zeros of a Cubic Polynomial

Zeros of a Polynomial

Zeros of a polynomial are those values of the variable(s) which when substituted in the polynomial make the polynomial equal to zero. They are also known as the roots of the polynomial.

Example of zeros of a polynomial:

1. For the polynomial 2x-4, the zero of this polynomial will be that value of x which will make this polynomial equal to zero, which is 2. (22)-4 which is equal to 0. Hence the zero or the root of the polynomial 2x-4 is 2. 
2. For the polynomial \(x^2+7x-18\), the zero or the root will be 2, because \((2)^2+72-18=4+14-18=0. \)

How to Find Zeros of Polynomials

The zeros of a polynomial are the x values for which the polynomial evaluates to zero. That is, these are the locations where the polynomial cuts or touches the x-axis.

A polynomial having a value of 0 for certain x is referred to as a zero polynomial. The degree of a polynomial is the maximum power of the variable x in the polynomial.

Following are the categories of polynomials depending upon their degree:

1. Linear Polynomial (Degree 1):

Form: ax + b, and a and b are real numbers and a ≠ 0

Example: 3x − 5 is a linear polynomial.

2. Quadratic Polynomial (Degree 2):

Form: ax² + bx + c, Where a, b, and c are real numbers and a ≠ 0

For Example,x² + 2x − 3 is a quadratic polynomial.

3. Cubic Polynomial (Degree 3):

Standard form: ax³ + bx² + cx + d, where a, b, c, and d are real numbers and a ≠ 0

For example, x³ + 2x² − x + 1 is a cubic polynomial.

Properties of Zero Polynomial

A zero polynomial is a polynomial in which all the coefficients are zero, and it is written simply as 0. It has some special properties that make it different from other polynomials. Here are the main properties explained in simple UK English:

All Coefficients Are Zero:
In a zero polynomial, every term has a coefficient of 0.
Example: 0x³ + 0x² + 0x + 0 = 0

Degree Is Undefined or -1:

In contrast to polynomials of other degrees, the degree of a zero polynomial cannot be defined or in certain situations is considered to be -1.

Because there is no term with a non-zero coefficient.

Additive Identity:

The zero polynomial is the additive identity in algebra.

Adding 0 to any polynomial doesn't change the polynomial.

Example: P(x) + 0 = P(x)

Multiplication Property:

Multiplying any polynomial by the zero polynomial results in the zero polynomial.

Example: P(x) · 0 = 0

No Roots in the Usual Sense:

The zero polynomial, being 0 for every value of x, has no distinct zeros as do ordinary polynomials.

Special Role in Equations:

When adding polynomials, the zero polynomial tends to show up when the terms cancel each other.

Difference Between zero polynomial and constant polynomial 

A zero polynomial is a polynomial where all coefficients are zero and its value is always 0, with an undefined degree. A constant polynomial is a non-zero number with degree 0 that does not depend on x. The main difference is that a zero polynomial is always 0, while a constant polynomial is a fixed non-zero value.

Solved Examples of Zero Polynomial

Example 1: If 2 is a zero polynomial p(x) = 4x2 + 2x – 5a, then value of a is

Given: 2 is a zero of p(x)

Calculation: p(2) = 0

Put 2 at the place of given in the polynomial,

⇒ 4(2)2 + 2 × 2 – 5a = 0

⇒ 4 × 4 + 4 – 5a = 0

⇒ 5a = 20

⇒ a = 4

∴ The value of a is 4.

Example 2: The zeroes of a polynomial are in arithmetic progression. The polynomial is x3 – 9x2 + 23x – 15. Find the smallest zero of the polynomial.

Solution: Let the zeroes of polynomial be a – d, a and a + d.

Sum of zeroes = -(coefficient of square of x)/ (coefficient of cube of x) = 9

So, a – d + a + a + d = 9

⇒ a = 3

Product of zeroes = -(constant)/(coefficient of cube of x) = 15

So, a(a – d)(a + d) = 15

⇒ (9 – d2) = 5

⇒ d = √4 = 2 or -2

Putting values of a and d, the zeroes are 1, 3 and 5. The smallest zero is 1.

Example 3: Sum of zeroes and reciprocal of its zeroes of a polynomial are 10 and 12.5 respectively. What is the product of zeroes of that polynomial?

Given: α + β = 10

1/α + 1/β = 12.5

Concept: Ifα and β are the zeroes of a polynomial then,

Sum of zeroes = α + β

Product of zeroes = αβ

Calculation:

α + β = 10 …………………(1)

1/α + 1/β = 12.5

⇒ (α + β)/αβ = 12.5 ……………………..(2)

From equation (1) and (2)

⇒10/ αβ = 12.5 = 25/2

⇒ αβ = 20/25

⇒ αβ = 4/5

∴ The product of zeroes of polynomial is 0.8

The correct option is 3 i.e. 0.8

Example 4: If the zeroes of polynomial ax2 + bx + c, are α and β then what will be the polynomial having zeroes α2 and β2.

Given: The polynomial is ax2 + bx + c

Zeroes of polynomial are α and β

Concept used: For a given polynomial ax2 + bx + c

Sum of zeroes = –b/a

Product of zeroes = c/a

And the polynomial can be written as x2 – Sx + P

Here, S is sum of zeroes and P is product of zeroes

Solution: Sum of zeroes = α + β = –b/a

Squaring both sides we’ll get

(α + β)2 = (b/a)2

⇒ α2 + β2 + 2αβ = b2/a2 —- (1)

Product of zeroes = α × β = c/a

⇒ αβ = c/a —- (2)

Solving (1) and (2) we’ll get

α2 + β2 + 2c/a = b2/a2

⇒ α2 + β2 = b2/a2 – 2c/a

⇒ α2 + β2 = (b2 – 2ac)/a2

As the zeroes of new polynomials are α2 and β2

Sum of zeroes (S) = (b2 – 2ac)/a2

Product of zeroes (P) = c2/a2

The new polynomial will be x2 – (b2 – 2ac)x/a2 + c2/a2

∴ The polynomial will be [a2x2 – (b2 – 2ac)x + c2]/a2

Example 5: If α and β are the two zeros of the polynomial 25x2 – 15x + 2, then what is a quadratic polynomial whose zeros are (2α)-1 and (2β)-1 ?

Concept: If α and β are the two zeros of the quadratic polynomial ax2 + bx + c, then:
ax2 + bx + c = (x – α)(x – β) = x2 – (α + β)x + αβ = 0

Sum of roots = α + β = −b/a

Product of roots = αβ = c/a.

Calculation:

25x2 – 15x + 2

⇒ 25x2 – 10x – 5x + 2

⇒ 5x(5x – 2) – 1(5x – 2)

⇒ (5x – 1)(5x – 2)

⇒ x = 1/5 and 2/5

⇒ α and β will be equal to 1/5 and 2/5.

If zeros are (2α)-1 and (2β)-1

Then the zeros of that polynomial are 5/2 and 5/4

∴ Polynomial = (x – 5/2)(x – 5/4) = (x2 – 15x/4 + 25/8) = (8x2 – 30x + 25)

Example 6: If -2, 3, and -5/2 are the zeroes of a cubic polynomial then which of the following options has the correct value of that cubic polynomial?

Given: Zeroes of a cubic polynomial = -2, 3, -5/2

Concept used: If a, b and c are the zeroes of a cubic polynomial, then the cubic polynomial is in the form: (x – a)(x – b)(x – c)

Calculation:

Finding the cubic polynomial using the above concept,

Cubic polynomial = (x + 2)(x – 3)(2x + 5)

⇒ (x2 + 2x – 3x – 6) × (2x + 5)

⇒ (x2 – x – 6) × (2x + 5)

⇒ 2x3 – 2x2 – 12x + 5x2 – 5x – 30

⇒ 2x3 + 3x2 – 17x – 30

∴ The cubic polynomial for given zeroes is 2x3 + 3x2 – 17x – 30

Example 7: If the sum of the zero of the equation is 2 and the product of the zero is -3 find the polynomial.

Let zeros of the equation is α and β respectively.

α + β = 2 and α β = -3

From the format of polynomial,

ax2 + bx + c = 0

By the relationship between zeros and coefficient,

α + β = -b/a

2 = -b/a

b = -2a.

α β = c/a

-3 = c/a

C = -3a.

ax2 – 2ax – 3a = 0

a(x2 – 2x – 3) = 0

x2 – 2x – 3 = 0

Example 8: The zeroes of the quadratic polynomial, the sum and product of whose zeroes are √2 and \(\frac{-3}{2}\), respectively, are:

Given: Sum of zeroes = √2

Product of zeroes = -3/2

Formula used:

α + β = -b/a

α × β = c/a

where,

α, β = Zeroes of quadratic polynomial

a = Coefficient of x2

b = Coefficient of x

c = Constant

Calculations:

We know the required polynomial is,

⇒ x2 – (α + β)x + αx = 0

⇒ x2 – √2x – (3/2) = 0

Multiplying the equation by 2,

⇒ 2x2 – 2√2x – 3 = 0

Finding the zeroes by splitting the middle term,

⇒ 2x2 – 2√2x – 3 = 0

⇒ 2x2 – 3√2x + √2x – 3 = 0

⇒ √2x(√2x – 3) + 1(√2x – 3) = 0

⇒ (√2x + 1)(√2x – 3) = 0

⇒ x = -1/√2 , x = 3/√2

∴ The zeroes of the quadratic equation are -1/√2 and 3/√2

Example 9: Find the zeros of the polynomial t2 – 15

Given: p(t) = t2 – 15

Calculation:

Zero of polynomial can be find out by putting p(t) = 0

⇒ t2 – 15 = 0

⇒ t2 = √15

∴ t = -√15 and √15 are zeroes of polynomial.

Example 10: Given that one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two roots is:

Given: One zero of the polynomial = 0

Formula used: Sum of the product of two zeroes = c/a

Calculation:

Let P(x) = ax3 + bx2 + cx + d

Let α, β and γ are the zeroes of cubic polynomial p(x),

where a = 0

Sum of the product of two zeroes = c/a

⇒ αβ + βγ + αγ = c/a

⇒ 0 × β + βγ + 0 × γ = c/a

⇒ 0 + βγ + 0 = c/a

⇒ βγ = c/a

∴ The product of the other two roots is c/a.

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