Chapter – 2

Polynomials

In this post we have given the detailed notes of class 10 Maths Chapter 2 (Polynomials) in English. These notes are useful for the students who are going to appear in class 10 board exams.

Chapter 2: Polynomials

What is a Polynomial?

An algebraic expression made up of variables, constants, coefficients of the variables, combined using addition, subtraction or multiplication, with only non-negative integer exponents, is called a polynomial.

Example: x² + 2x + 1 is a polynomial expression.

• 2x⁵ + 4xy³ + 6x²
• 4y³ + y² + yz
• 3x + x² − x⁴
• 5x⁶y + 6px²yx² − 8ax

A polynomial in one variable x of degree n is expressed as:

p(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀

where a_n ≠ 0, and a_n, a_n−1, …, a₁, a₀ are constants (real numbers).

Degree of a Polynomial

The highest power of the variable in a polynomial is called the degree of the polynomial. If more than one variable is present, the highest sum of powers of the variables in any single term is taken as the degree of the polynomial.

Example: In the polynomial 2y² − 3y + 4, the highest power of the variable y is 2, so the degree of the polynomial is 2.

Linear Polynomial

A polynomial of degree 1 is called a linear polynomial. For example, 2x − 3, √3x + 5, y + √2, etc.

The most general form of a linear polynomial is ax + b, where a and b are real numbers and a ≠ 0.

Quadratic Polynomial

A polynomial of degree 2 is called a quadratic polynomial. The word “quadratic” comes from the word “quadrate”, which means “square”.

For example, 2x² + 3x − 2/5, y² − 2, etc.

More generally, any quadratic polynomial in x is of the form ax² + bx + c, where a, b, c are real numbers and a ≠ 0.

Cubic Polynomial

A polynomial of degree 3 is called a cubic polynomial. Some examples of cubic polynomials are:

2 − x³, x³, x³ − x² + 3, etc.

The most general form of a cubic polynomial is ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.

Algebraic Polynomial

An expression obtained by combining variable and constant terms is called an algebraic polynomial. For example, x + 2, x + 6, y − 4, 64 + a.

Algebraic polynomials are of two types:

1. Constant Polynomial

A term of a polynomial whose value always remains fixed is called a constant polynomial.

Examples: 2 and 5 are constant polynomials because their value is always fixed.

Note:

• Constant polynomials can be real or imaginary numbers.
• The degree of a constant polynomial is zero.

2. Variable Polynomial

A term of a polynomial whose value keeps changing is called a variable polynomial.

Examples: x² + 4x + 2, 2x² + 4x + 8.

Note: A variable polynomial can never be imaginary.

Zeroes of a Polynomial

A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

In general, if p(x) = ax + b has a zero k, then p(k) = ak + b = 0, i.e. k = −b/a. So the zero of a linear polynomial ax + b is −b/a = −(constant term)/(coefficient of x).

Geometrical Meaning of the Zeroes of a Polynomial

The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

• A linear polynomial has at most one zero — the graph of y = ax + b is a straight line that crosses the x-axis at exactly one point.
• A quadratic polynomial can have at most two zeroes — the graph of y = ax² + bx + c is a parabola that can cross the x-axis at 0, 1, or 2 points.
• A cubic polynomial can have at most three zeroes.

In general, a polynomial of degree n can have at most n zeroes.

Relationship Between Zeroes and Coefficients of a Quadratic Polynomial

Let us try to understand the relationship between the zeroes and coefficients of a polynomial through an example.

Consider the quadratic polynomial p(x) = 2x² − 8x + 6. Here we need to split the middle term “−8x” into two terms whose product is 6x × 2x = 12x².

So we can write:

2x² − 8x + 6 = 2x² − 2x − 6x + 6

= 2x(x − 1) − 6(x − 1) = (x − 1)(2x − 6)

= 2(x − 1)(x − 3)

So the value of p(x) = 2x² − 8x + 6 is zero when x = 1 or x = 3.

So, the zeroes of the quadratic polynomial p(x) = 2x² − 8x + 6 are 1 and 3.

Sum of Zeroes

= 1 + 3 = 4 = −(−8)/2 = −(coefficient of x)/(coefficient of x²)

Product of Zeroes

= 1 × 3 = 3 = 6/2 = (constant term)/(coefficient of x²)

In general, if α, β are the zeroes of a quadratic polynomial p(x) = ax² + bx + c, a ≠ 0, then (x − α) and (x − β) are factors of p(x).

ax² + bx + c = k(x − α)(x − β), where k is a constant

= k[x² − (α + β)x + αβ]

= kx² − k(α + β)x + kαβ

Comparing the coefficients of x², x and the constant terms on both sides, we get:

a = k, b = −k(α + β), and c = kαβ

This gives:

α + β = −b/a = −(coefficient of x)/(coefficient of x²)

αβ = c/a = (constant term)/(coefficient of x²)

Finding a Quadratic Polynomial from the Sum and Product of its Zeroes

Example: Find a quadratic polynomial, the sum and product of whose zeroes are −3 and 2 respectively.

Solution: Let the quadratic polynomial be ax² + bx + c, and its zeroes be α, β.

We have α + β = −b/a = −3

αβ = c/a = 2

If a = 1, then b = 3 and c = 2.

So one quadratic polynomial satisfying the given conditions is x² + 3x + 2.

Important Facts

• A quadratic polynomial can have at most two zeroes, and a cubic polynomial can have at most three zeroes.
• If α, β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, a ≠ 0, then α + β = −b/a and αβ = c/a.

Addition, Subtraction, Multiplication and Division of Polynomials

Addition of Polynomials

When we add two or more polynomials, only like terms (terms with the same variable and same power) are added. Unlike terms remain unchanged. In addition, the degree of the resulting polynomial stays the same.

Q1. Add the polynomials 5x² + 4x + 2 and 8x² + 2x + 5.

Solution: 5x² + 4x + 2 + 8x² + 2x + 5

= (5x² + 8x²) + (4x + 2x) + (2 + 5)

= 13x² + 6x + 7

Ans. 13x² + 6x + 7

Q2. Add the polynomials 3a² + 5ab + 2 and 7a² + 6 + 9ab.

Solution: 3a² + 5ab + 2 + 7a² + 6 + 9ab

= (3a² + 7a²) + (5ab + 9ab) + (2 + 6)

= 10a² + 14ab + 8

Ans. 10a² + 14ab + 8

Subtraction of Polynomials

Subtraction of polynomials works the same way as addition — like terms are subtracted, and unlike terms remain unchanged. Here too, the degree of the resulting polynomial stays the same.

Q1. Subtract 3xy + 2xy² + x²y + 2y³ from 5xy + 8xy² + 6x²y + 8y³.

Solution: (5xy + 8xy² + 6x²y + 8y³) − (3xy + 2xy² + x²y + 2y³)

= (8y³ − 2y³) + (6x²y − x²y) + (8xy² − 2xy²) + (5xy − 3xy)

= 6y³ + 5x²y + 6xy² + 2xy

Ans. 6y³ + 5x²y + 6xy² + 2xy

Multiplication of Polynomials

When two or more polynomials are multiplied, the result is always a polynomial of higher degree, unless one or both of the polynomials being multiplied is a constant polynomial, in which case the degree stays the same. While multiplying, the powers of like variables are added using the laws of exponents.

Q1. Multiply 2x × 4y.

Solution: 2x × 4y = (2 × 4) × (x × y) = 8xy

Ans. 8xy

Q2. Multiply 5a × 8b.

Solution: 5a × 8b = (5 × 8) × (a × b) = 40ab

Ans. 40ab

Q3. Multiply 3p²q² × 12p³q³.

Solution: 3p²q² × 12p³q³ = (3 × 12) × (p² × p³ × q² × q³) = 36p⁵q⁵

Ans. 36p⁵q⁵

Division of Polynomials

When a polynomial is divided, the result is always a polynomial of lower degree, unless one of the polynomials is a constant polynomial, in which case the degree stays the same.

Q1. Divide 6a² ÷ 3a.

Solution: 6a² ÷ 3a = (3 × 2 × a × a)/(3 × a) = 2a

Ans. 2a

Q2. Divide (2xy + 6x) ÷ 2x.

Solution: (2xy + 6x)/2x = 2x(y + 3)/2x = y + 3

Ans. y + 3

Zeroes of a Cubic Polynomial

If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d, then it can be shown that:

α + β + γ = −b/a

αβ + βγ + γα = c/a

αβγ = −d/a

Example of a Cubic Polynomial

Verify that 3, −1, and −1/3 are the zeroes of the cubic polynomial p(x) = 3x³ − 5x² − 11x − 3, and then verify the relationship between the zeroes and the coefficients.

Solution: Comparing the given polynomial with ax³ + bx² + cx + d, we get a = 3, b = −5, c = −11, d = −3.

Substituting the zeroes one by one:

p(3) = 3(3)³ − 5(3)² − 11(3) − 3 = 81 − 45 − 33 − 3 = 0

p(−1) = 3(−1)³ − 5(−1)² − 11(−1) − 3 = −3 − 5 + 11 − 3 = 0

p(−1/3) = 3(−1/3)³ − 5(−1/3)² − 11(−1/3) − 3 = −1/9 − 5/9 + 11/3 − 3 = 0

So, 3, −1 and −1/3 are the zeroes of 3x³ − 5x² − 11x − 3.

Taking α = 3, β = −1, γ = −1/3:

α + β + γ = 3 + (−1) + (−1/3) = 5/3 = −b/a = −(−5)/3 ✓

αβ + βγ + γα = (3)(−1) + (−1)(−1/3) + (−1/3)(3) = −3 + 1/3 − 1 = −11/3 = c/a = −11/3 ✓

αβγ = (3)(−1)(−1/3) = 1 = −(−3)/3 = −d/a ✓

Key Facts

If α, β, γ are the zeroes of the cubic polynomial p(x) = ax³ + bx² + cx + d, then α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a.

Division Algorithm for Polynomials

You know that a cubic polynomial can have at most three zeroes. But if you are given only one zero, can you find the other two? For example, consider the cubic polynomial x³ − 3x² − x + 3. If we know that one of its zeroes is 1, then x − 1 is a factor of x³ − 3x² − x + 3. So dividing x³ − 3x² − x + 3 by x − 1, we get the quotient x² − 2x − 3.

The quadratic polynomial so obtained can be factorised by splitting the middle term:

x² − 2x − 3 = x² − 3x + x − 3

= x(x − 3) + 1(x − 3) = (x − 3)(x + 1)

So all the zeroes of the cubic polynomial are 1, −1 and 3.

Steps of the Division Algorithm

The systematic steps to divide one polynomial by another are worked through here with an example. If p(x) and g(x) are any two polynomials with g(x) ≠ 0, we can find polynomials q(x) and r(x) such that:

p(x) = g(x) × q(x) + r(x)

This result is called the Division Algorithm for polynomials.

Example: Divide 2x² + 3x + 1 by x + 2.

Solution: Note that we stop the division process when either the remainder is zero, or its degree is less than the degree of the divisor.

Dividend = Divisor × Quotient + Remainder

2x² + 3x + 1 = (x + 2) × (2x − 1) + 3

Points to Remember

• This process can also be used to divide a polynomial by a quadratic polynomial.
• According to the Division Algorithm, for a given polynomial p(x) and any non-zero polynomial g(x), there exist polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

Key Points to Remember

• A polynomial is an algebraic expression with non-negative integer powers of the variable(s).
• Degree = highest power of the variable in the polynomial: degree 1 = linear, degree 2 = quadratic, degree 3 = cubic.
• The zeroes of p(x) are the x-coordinates where the graph of y = p(x) crosses the x-axis.
• A linear polynomial has at most 1 zero, a quadratic at most 2, and a cubic at most 3.
• For a quadratic polynomial ax² + bx + c: sum of zeroes α + β = −b/a; product of zeroes αβ = c/a.
• For a cubic polynomial ax³ + bx² + cx + d: α + β + γ = −b/a; αβ + βγ + γα = c/a; αβγ = −d/a.
• Division Algorithm: p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

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