## CBSE Class 10 Maths Notes Chapter 2 Polynomials

- A polynomial is an expression that consists of variables and coefficients and involves only the operations of addition, subtraction, and multiplication.
- The degree of a polynomial is the highest power of the variable present in the polynomial.
- A polynomial of degree 0 is called a constant polynomial.
- A polynomial of degree 1 is called a linear polynomial.
- A polynomial of degree 2 is called a quadratic polynomial.
- A polynomial of degree 3 is called a cubic polynomial.
- The standard form of a quadratic polynomial is ax² + bx + c, where a, b, and c are constants and a ≠ 0.
- The factor theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is f(a).
- The remainder theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is zero if and only if f(a) = 0.
- A polynomial with real coefficients can be factored into linear and quadratic factors with real coefficients.
- The zeroes of a polynomial are the values of the variable for which the polynomial becomes zero.
- The Fundamental Theorem of Algebra states that every polynomial equation of degree n has n complex roots, including repeated roots.
- The sum of the zeroes of a polynomial is equal to the negative coefficient of the term of the second highest degree, divided by the coefficient of the term of the highest degree.
- The product of the zeroes of a polynomial is equal to the constant term divided by the coefficient of the term of the highest degree.

## CBSE Class 10 Maths Important Questions Chapter 2 Polynomials

- State the factor theorem.
- State the remainder theorem.
- Find the zeroes of the polynomial x^3 – 3x^2 – 4x + 12, if two of its zeroes are 2 and -2.
- Factorize the polynomial 6x^2 – 5x – 6.
- Find a polynomial of degree 3 with zeroes 1, 2 and -3.
- If (x – 2) is a factor of the polynomial 2x^3 + ax^2 + bx – 12, find the value of a, b, and the third factor.
- If the zeroes of the polynomial ax^2 + bx + c are equal, find the value of b^2 – 4ac.
- Find the value of k for which x – 1 is a factor of the polynomial 3x^3 – 4kx^2 + 3x + 2k.
- If the zeroes of the polynomial x^3 – 2x^2 – 5x + 6 are a, b and c, find the value of a + b + c.
- Factorize the polynomial x^4 – 4x^3 + 6x^2 – 4x + 1.

## CBSE Class 10 Maths Important Questions Answers Chapter 2 Polynomials

- The factor theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is f(a).
- The remainder theorem states that if a polynomial f(x) is divided by (x – a), then the remainder is zero if and only if f(a) = 0.
- Using the given zeroes, the polynomial can be written as (x-2)(x+2)(x-3). Thus, the zeroes of the polynomial are 2, -2 and 3.
- The polynomial can be factorized as (2x+3)(3x-2).
- A polynomial of degree 3 with zeroes 1, 2 and -3 is (x-1)(x-2)(x+3).
- Since (x-2) is a factor of the given polynomial, f(2) = 0. Therefore, we have 16a + 4b – 12 = 0. Also, the third factor is (2x + p), where p is a constant. Using the given polynomial, we have (2x+p)(x-2)^2 = 2x^3 + ax^2 + bx – 12. Expanding this equation, we get p = -6, a = 12, and b = -25.
- If the zeroes of the polynomial are equal, then the discriminant b^2 – 4ac is equal to zero. Therefore, we have b^2 – 4ac = 0.
- If x – 1 is a factor of the polynomial, then f(1) = 0. Using the given polynomial, we have 3 – 4k + 3 + 2k = 0. Therefore, we have k = 1/2.
- Using the given zeroes, the polynomial can be written as (x-a)(x-b)(x-c) = x^3 – (a+b+c)x^2 + (ab+bc+ca)x – abc. Equating the coefficients with the given polynomial, we get a + b + c = 2, ab + bc + ca = -5, and abc = -6. Solving these equations, we get a = 1, b = -2, and c = -3. Therefore, a + b + c = -4.
- The given polynomial can be written as (x-1)^4.

**CBSE Class 10 Maths Important Questions Answers MCQs Chapter 2 Polynomials**

The degree of the polynomial 5x^4 – 3x^2 + 2 is:

a) 4

b) 3

c) 2

d) 1

Answer: a) 4

Which of the following is not a polynomial?

a) x^3 + 2x^2 – x + 5

b) 3x – 2

c) 5x^2 – 7x + 1/x

d) 2x^2 + 5x – 1

Answer: c) 5x^2 – 7x + 1/x

If the zeroes of the polynomial x^3 – 3x^2 + x + 2 are a, b and c, then their sum is:

a) -1

b) 1

c) 2

d) 3

Answer: b) 1

If (x – 2) is a factor of the polynomial x^3 + ax^2 + bx + 6, then the value of b is:

a) -6

b) 6

c) 2

d) -2

Answer: a) -6

The polynomial 3x^2 – 2x + 1 is a:

a) Quadratic polynomial

b) Linear polynomial

c) Cubic polynomial

d) Zero polynomial

Answer: a) Quadratic polynomial

If (x – 1) is a factor of the polynomial 2x^3 + 3x^2 – 5x + k, then the value of k is:

a) 1

b) -1

c) 0

d) 2

Answer: b) -1

The zeroes of the polynomial x^2 – 4x + 4 are:

a) 2, 2

b) 2, -2

c) 4, -4

d) -2, -2

Answer: a) 2, 2

If a polynomial has degree 2, then it can have:

a) 0 zeroes

b) 1 zero

c) 2 zeroes

d) 3 zeroes

Answer: c) 2 zeroes

The zeroes of the polynomial 2x^2 + 5x – 3 are:

a) -1/2, 3/2

b) 1/2, -3/2

c) -3/2, -1/2

d) 3/2, -1/2

Answer: a) -1/2, 3/2

If a polynomial has degree 3, then it can have:

a) 1 zero

b) 2 zeroes

c) 3 zeroes

d) 4 zeroes

Answer: c) 3 zeroes

## CBSE Class 10 Maths Notes Chapter 2 Polynomials

- “Polynomial” comes from the word ‘Poly’ (Meaning Many) and ‘nomial’ (in this case meaning Term)-so it means many terms.
- A polynomial is made up of terms that are only added, subtracted or multiplied.
- A quadratic polynomial in x with real coefficients is of the form ax² + bx + c, where a, b, c are real numbers with a ≠ 0.
- Degree – The highest exponent of the variable in the polynomial is called the degree of polynomial. Example: 3x
^{3}+ 4, here degree = 3. - Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomial respectively.
- A polynomial can have terms which have Constants like 3, -20, etc., Variables like x and y and Exponents like 2 in y².
- These can be combined using addition, subtraction and multiplication but NOT DIVISION.
- 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.

If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then

sumofzeros,α+β=−ba=−coefficientofxcoefficientofx2

productofzeros,αβ=ca=constanttermcoefficientofx2

If α, β, γ are the zeroes of the cubic polynomial ax^{3} + bx^{2} + cx + d = 0, then

α+β+γ=−ba=−coefficientofx2coefficientofx3

αβ+βγ+γα=ca=coefficientofxcoefficientofx3

αβγ=−da=−constanttermcoefficientofx3

Zeroes (α, β, γ) follow the rules of algebraic identities, i.e.,

(α + β)² = α² + β² + 2αβ

∴(α² + β²) = (α + β)² – 2αβ

**DIVISION ALGORITHM:**

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then

p(x) = g(x) × q(x) + r(x)

Dividend = Divisor x Quotient + Remainder

**Remember this!**

- If r (x) = 0, then g (x) is a factor of p (x).
- If r (x) ≠ 0, then we can subtract r (x) from p (x) and then the new polynomial formed is a factor of g(x) and q(x).

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