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101380_6Class_9_chapter_2_Polynomials_1.pdf NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

Polynomials

Exercise 2.1

Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (i) ૝ܠ૛െ૜ܠ

Answer: Ͷଶെ͵൅͹ can also be written as Ͷଶെ͵ଵ൅͹଴ .

We can call the expression Ͷଶെ͵൅͹ a polynomial in one variable because x

is the only variable in the equation and the powers of x (i.e., 2, 1, and 0) are all whole numbers. (ii) ܡ Answer: ଶ൅ξʹ can also be written as ଶ൅ξʹ଴ . We can call the expression ଶ൅ξʹ a polynomial in one variable because y is the only variable in the equation and the powers of x (i.e., 2 and 0) are all whole numbers. (iii) ૜ξܜ൅ܜ Answer: ͵ξ൅ξʹ can also be written as ͵ భ మ൅ξʹଵ . Despite the fact that t is the only variable in the equation, the powers of t (i.e.,1/2) is not whole numbers. As a result, we can say that the equation ͵ξ൅

ξʹ is not a one-variable polynomial.

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (iv) ܡ ܡ

Answer: ൅ଶ

୷ can also be written as ଵ൅ʹିଵ . Despite the fact that y is the only variable in the equation, the powers of t (i.e., -1) is not whole numbers. As a result, we can say that the equation ൅ଶ ୷ is not a one-variable polynomial. (v) ܠ૚૙൅ܡ૜൅ܜ Answer: In this case, the equation is ଵ଴൅ଷ൅ହ଴ Despite the fact that the powers 10, 3, and 50 are all whole numbers, the expression has three variables. As a result, it is not a one-variable polynomial. Q2. Write the coefficients of x2 in each of the following: (i) ૛൅ܠ૛൅ܠ Answer: ʹ൅ଶ൅ can also be written as ʹ൅ሺͳሻଶ൅ . The number that multiplies the variable is known as the coefficient.

The variable x2 is multiplied by 1 in this case.

In ʹ൅ଶ൅ , the coefficients of x2 is 1. (ii) ૛െܠ૛൅ܠ

Answer: ʹെଶ൅ଷ

can also be written as ʹ൅ሺെͳሻଶ൅ଷ . The number (together with its sign, i.e., ʹ or +) that multiplies the variable is known as the coefficient. The variable x2 is multiplied by -1 in this case. NCERT Solutions for Class 9 Maths Chapter 2- Polynomials In ʹെଶ൅ଷ , the coefficients of x2 is -1. (iii) ቀ஠ ૛ቁܠ૛൅ܠ

Answer: ቀ஠

ଶቁଶ൅ is a form of the equation ቀ஠ ଶቁଶ൅ . The number (together with its sign, i.e., ʹ or +) that multiplies the variable is known as the coefficient. The variable x2 is multiplied by the number ஠ ଶ in this case.

The x2 coefficients in ቀ஠

ଶቁଶ൅ is ஠ ଶ . (iv) ξʹെͳ

Answer: Since Ͳଶ is 0, the equation ξʹെͳ can be written as Ͳଶ൅ξʹെ

ͳ .

The number (together with its sign, i.e., ʹ or +) that multiplies the variable is known as the coefficient. The variable x2 is multiplied by 0 in this case. In ξʹെͳ , the x2 coefficients in

ξʹെͳ is 0.

Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100. Answer: Binomial of degree 35: A binomial of degree 35 is a polynomial with two terms and the greatest degree 35.

Example: ͳͲଷହ൅ͳͲ .

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials A polynomial with one term and the greater degree 100 is referred to as a monomial of degree 100. Example: ͳͷଵ଴଴൅͵ . Q4. Write the degree of each of the following polynomials: (i) ૞ܠ૜൅૝ܠ૛൅ૠܠ Answer: The degree of a polynomial is the largest power of the variable in the polynomial. ͷଷ൅Ͷଶ൅͹ൌͷଷ൅Ͷଶ൅͹ଵ x has the following powers: 3, 2, 1

Because 3 is the largest power of x in the equation, the degree of ͷଷ൅Ͷଶ൅͹

is 3. (ii) ૝െܡ Answer: The degree of a polynomial is the largest power of the variable in the polynomial. ͶെଶൌͶ଴െଶ The highest power of the variable y is 2. Therefore, the degree of the given polynomial is 2. (iii) ૞ܜ Answer: The degree of a polynomial is the largest power of the variable in the polynomial. ͷଵെξ͹ The highest power of the variable t is 1. Therefore, the degree of the polynomial is 1. NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (iv) ͵ Answer: The degree of a polynomial is the largest power of the variable in the polynomial. Since there are no variables, we can assume that the power of any variable would be 0 here. Therefore, the degree of the polynomial is 0. Q5. Classify the following as linear, quadratic and cubic polynomials: (i) ܠ૛൅ܠ Answer: The degree of the polynomial is 2. Therefore, it is a quadratic polynomial. (ii) െଷ Answer: The degree of the polynomial is 3. Therefore, it is a cubic polynomial. (iii) ൅ଶ൅Ͷ Answer: The degree of the polynomial is 2. Therefore, it is a quadratic polynomial. (iv) ͳ൅ Answer: The degree of the polynomial is 1. Therefore, it is a linear polynomial. (v) ͵ Answer: The degree of the polynomial is 1. Therefore, it is a linear polynomial. (vi) ଶ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials Answer: The degree of the polynomial is 2. Therefore, it is a quadratic polynomial. (vii) ͹ଷ Answer: The degree of the polynomial is 3. Therefore, it is a cubic polynomial.

Exercise 2.2:

Q1. Find the value of the polynomial ૞ܠെ૝ܠ (i) x = 0 (ii) x = ʹ 1 (iii) x = 2

Answer:

(i) Let us assume that ሺሻൌͷെͶଶ൅͵

Substituting ൌͲ , we get:

ሺͲሻൌͷሺͲሻെͶሺͲሻଶ൅͵ ൌ͵ (ii) Substituting ൌെͳ , we get:

ሺെͳሻൌͷሺെͳሻെͶሺെͳሻଶ൅͵

ൌെ͸ (iii) Substituting ൌʹ , we get: ሺെͳሻൌͷሺʹሻെͶሺʹሻଶ൅͵ ൌെ͵ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials Q2. Find p(0), p(1) and p(2) for each of the following polynomials: (i) ܘሺܡሻൌܡ૛െܡ

Answer:

ሺͲሻൌሺͲሻଶെሺͲሻ൅ͳൌͳ ሺͳሻൌሺͳሻଶെሺͳሻ൅ͳൌͳ ሺʹሻൌሺʹሻଶെሺʹሻ൅ͳൌ͵ (ii) ܘሺܜሻൌ૛൅ܜ൅૛ܜ૛െܜ

Answer:

ሺͲሻൌʹ൅ሺͲሻ൅ʹሺͲሻଶെሺͲሻଷൌʹ

ሺͳሻൌʹ൅ሺͳሻ൅ʹሺͳሻଶെሺͳሻଷൌͶ

ሺͲሻൌʹ൅ሺʹሻ൅ʹሺʹሻଶെሺʹሻଷൌͶ

(iii) ܘሺܠሻൌܠ

Answer:

ሺͲሻൌሺͲሻଷൌͲ ሺͳሻൌሺͳሻଷൌͳ ሺʹሻൌሺʹሻଷൌͺ (iv) ܘሺܠሻൌሺܠെ૚ሻሺܠ

Answer:

ሺͲሻൌ൫ሺͲሻെͳ൯൫ሺͲሻ൅ͳ൯ൌെͳ

ሺͳሻൌ൫ሺͳሻെͳ൯൫ሺͳሻ൅ͳ൯ൌͲ

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

ሺʹሻൌ൫ሺʹሻെͳ൯൫ሺʹሻ൅ͳ൯ൌ͵

Q3. Verify whether the following are zeroes of the polynomial, indicated against them. (i) ܘሺܠሻൌ૜ܠ൅૚ǡܠ Answer: For the value of ൌെͳȀ͵ , the output of the polynomial will be: ቀെଵ ଷቁൌ͵ቀെଵ ଷቁ൅ͳൌͲ Therefore, the value satisfies the polynomial, thus it is a zero of the polynomial. (ii) ܘሺܠሻൌ૞ܠെૈǡܠ Answer: For the value of ൌͶȀͷ , the output of the polynomial will be: ሺͶȀͷሻൌͷቀସ ହቁെɎൌͶെɎ Therefore, it does not satisfy the polynomial and is not a zero of it. (iii) ܘሺܠሻൌܠ૛െ૚ǡܠ Answer: For the value of ൌͳ, the output of the polynomial will be: ሺͳሻൌሺͳሻଶെͳൌͲ For the value of ൌെͳ , the output of the polynomial will be: ሺെͳሻൌሺെͳሻଶെͳൌͲ Therefore, ͳെͳ both are the zeroes of the polynomial. (iv) ܘሺܠሻൌሺܠ൅૚ሻሺܠെ૛ሻǡܠ Answer: For the value of ൌെͳ, the output of the polynomial will be: NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

ሺെͳሻൌሺെͳ൅ͳሻሺെͳെʹሻൌͲ

For the value of ൌʹ, the output of the polynomial will be: ሺʹሻൌሺʹ൅ͳሻሺʹെʹሻൌͲ Therefore, -1 and 2 both are the zeroes of the polynomial. (v) ܘሺܠሻൌܠ૛ǡܠ Answer: For the value of ൌͲ, the output of the polynomial will be: ሺͲሻൌͲଶൌͲ

Therefore, 0 is the zero of the polynomial.

(vi) ܘሺܠሻൌܠܔ൅ܕǡܠൌെܕȀܔ Answer: For the value of ൌെȀ , the output of the polynomial will be: ሺെȀሻൌቀെ୫ ୪ቁ൅ൌͲ

Therefore, െ୫

୪ is the zero of the polynomial. (vii) ܘሺܠሻൌ૜ܠ૛െ૚ǡܠ

ξ૜ǡ૛

ξ૜

Answer: For the value of ൌെଵ

ξଷ , the output of the polynomial will be: ቀെଵ

ξଷቁൌ͵ቀെଵ

ξଷቁ

ଶെͳൌͲ For the value of ൌʹȀξ͵ , the output of the polynomial will be: ቀଶ

ξଷቁൌ͵ቀଶ

ξଷቁ

ଶെͳൌ͵

Therefore, െଵ

ξଷ is the zero of the polynomial.

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (viii) ܘሺܠሻൌ૛ܠ൅૚ǡܠ ૛ Answer: For the value of ൌͳȀʹ , the output of the polynomial will be: ቀଵ ଶቁൌʹቀଵ ଶቁ൅ͳൌʹ

Therefore, ଵ

ଶ is not the zero of the polynomial. Q4. Find the zero of the polynomials in each of the following cases: (i) ܘሺܠሻൌܠ

Answer:

ሺሻൌ൅ͷ ൌ൐൅ͷൌͲ ൌ൐ൌെͷ

Therefore, -5 is a zero of the given polynomial.

(ii) ܘሺܠሻൌܠ

Answer:

ሺሻൌെͷ ൌ൐െͷൌͲ ൌ൐ൌͷ

Therefore, 5 is a zero of the given polynomial.

(iii) ܘሺܠሻൌ૛ܠ

Answer:

ሺሻൌʹ൅ͷ ൌ൐ʹ൅ͷൌͲ ൌ൐ʹൌെͷ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ൌ൐ൌെͷȀʹ Therefore, -5/2 is a zero of the given polynomial. (iv) ܘሺܠሻൌ૜ܠ

Answer:

ሺሻൌ͵Ȃʹ ൌ൐͵െʹൌͲ ൌ൐͵ൌʹ ൌ൐ൌʹȀ͵ Therefore, -2/3 is a zero of the given polynomial. (v) ܘሺܠሻൌ૜ܠ

Answer:

ሺሻൌ͵ ൌ൐͵ൌͲ ൌ൐ൌͲ

Therefore, 0 is a zero of the given polynomial.

(vi) ܘሺܠሻൌܠ܉ǡ܉

Answer:

ሺሻൌ ൌ൐ൌͲ ൌ൐ൌͲ

Therefore, 0 is a zero of the given polynomial.

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

(vii) ܘሺܠሻൌܠ܋൅܌ǡ܋്૙ǡ܋ǡܛܚ܍܊ܕܝܖܔ܉܍ܚ܍ܚ܉܌

Answer:

ሺሻൌ൅ ൌ൐൅ൌͲ ൌ൐ൌെȀ Therefore, -d/c is a zero of the given polynomial.

Exercise 2.3:

Q1. Find the remainder when ܠ૜൅૜ܠ૛൅૜ܠ (i) ܠ

Answer:

൅ͳൌͲ ֜ ׵

ሺെͳሻൌሺെͳሻଷ൅͵ሺെͳሻଶ൅͵ሺെͳሻ൅ͳ

ൌെͳ൅͵െ͵൅ͳ ൌͲ (ii) ܠ ૛

Answer:

െͳȀʹൌͲ ֜ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ׵ ቀଵ ଶቁൌቀଵ ଶቁ ଷ൅͵ቀଵ ଶቁ ଶ൅͵ቀଵ ଶቁ൅ͳ ൌቀଵ ଼ቁ൅ቀଷ ସቁ൅ቀଷ ଶቁ൅ͳ ൌʹ͹Ȁͺ (iii) ܠ

Answer:

ൌͲ ׵

ሺͲሻൌሺͲሻଷ൅͵ሺͲሻଶ൅͵ሺͲሻ൅ͳ

ൌͳ (iv) ܠ

Answer:

൅ɎൌͲ ֜ ׵

ሺͲሻൌሺെɎሻଷ൅͵ሺെɎሻଶ൅͵ሺെɎሻ൅ͳ

ൌെɎଷ൅͵Ɏଶെ͵Ɏ൅ͳ (v) ૞൅૛ܠ

Answer:

ͷ൅ʹൌͲ ֜ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ֜ ׵ ቀെହ ଶቁ ଷ൅͵ቀെହ ଶቁ ଶ൅͵ቀെହ ଶቁ൅ͳ ൌቀെଵଶହ ଼ቁ൅ቀ଻ହ ସቁെቀଵହ ଶቁ൅ͳ ൌെଶ଻ ଼ Q2. Find the remainder when ܠ૜െܠ܉૛൅૟ܠെ܉

Answer:

Let us assume

ሺሻൌଷെଶ൅͸െ െൌͲ ൌ

The Remainder will be:

ሺሻൌሺሻଷെሺଶሻ൅͸ሺሻെ ൌଷെଷ൅͸െ ൌͷ Q3. Check whether 7+3x is a factor of ૜ܠ૜൅ૠܠ

Answer:

͹൅͵ൌͲ ֜ ֜ ׵ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ͵ቀെ଻ ଷቁ ଷ൅͹ቀെ଻ ଷቁ ൌെቀଷସଷ ଽቁ൅ቀെସଽ ଷቁ ൌିଷସଷିሺସଽሻଷ ଽ ൌିଷସଷିଵସ଻ ଽ ൌെସଽ଴ ଽ്Ͳ Therefore, ͹൅͵ is not a factor of ͵ଷ൅͹ .

Exercise 2.4

Q1. Determine which of the following polynomials has (x + 1) a factor: (i) ܠ૜൅ܠ૛൅ܠ

Answer:

Assuming a function:

ሺሻൌଷ൅ଶ൅൅ͳ ൅ͳൌͲ =>ൌെͳ Substituting ൌെͳ in the polynomial.

ሺെͳሻൌሺെͳሻଷ൅ሺെͳሻଶ൅ሺെͳሻ൅ͳ

ൌെͳ൅ͳെͳ൅ͳ ൌͲ Therefore, according to factor theorem, x+1 is a factor of the given polynomial. (ii) ܠ૝൅ܠ૜൅ܠ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

Answer:

Assuming a function:

ሺሻൌସ൅ଷ൅ଶ൅൅ͳ ൅ͳൌͲ =>ൌെͳ Substituting ൌെͳ in the polynomial.

ሺെͳሻൌሺെͳሻସ൅ሺെͳሻଷ൅ሺെͳሻଶ൅ሺെͳሻ൅ͳ

ൌͳെͳ൅ͳെͳ൅ͳ ൌͳ്Ͳ Therefore, according to the factor theorem, x+1 is not a factor of the given polynomial. (iii) ܠ૝൅૜ܠ૜൅૜ܠ૛൅ܠ

Answer:

Assuming a function:

ሺሻൌସ൅͵ଷ൅͵ଶ൅൅ͳ ൅ͳൌͲ =>ൌെͳ Substituting ൌെͳ in the polynomial.

ሺെͳሻൌሺെͳሻସ൅͵ሺെͳሻଷ൅͵ሺെͳሻଶ൅ሺെͳሻ൅ͳ

ൌͳെ͵൅͵െͳ൅ͳ ൌͳ്Ͳ Therefore, according to the factor theorem, x+1 is not a factor of the given polynomial. (iv) ܠ૜െܠ૛െ൫૛൅ξ૛൯ܠ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

Answer:

Assuming a function:

ሺሻൌଷ൅ଶെሺʹ൅ξʹሻ൅ξʹ ൅ͳൌͲ =>ൌെͳ Substituting ൌെͳ in the polynomial.

ሺെͳሻൌሺെͳሻଷȂሺെͳሻʹȂ൫ʹ൅ξʹ൯ሺെͳሻ൅ξʹ

ൌെͳെͳ൅ʹ൅ξʹ൅ξʹ ൌʹξʹ്Ͳ Therefore, according to the factor theorem, x+1 is not a factor of the given polynomial. Q2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:

(i) ܘሺܠሻൌ૛ܠ૜൅ܠ૛Ȃ૛ܠȂ૚ǡ܏ሺܠሻൌܠ

Answer:

Firstly, we will calculate the zero of g(x) then we will substitute the zero of g(x) in p(x) to see if g(x) is a factor of p(x) or not. ሺሻൌͲ ൅ͳൌͲ ൌെͳ

Substituting the zero of g(x) in p(x).

ሺെͳሻൌʹሺെͳሻଷ൅ሺെͳሻଶȂʹሺെͳሻȂͳ

ൌെʹ൅ͳ൅ʹെͳ ൌͲ Therefore, according to the factor theorem g(x) is a factor of p(x). NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

(ii) ܘሺܠሻൌܠ૜൅૜ܠ૛൅૜ܠ൅૚ǡ܏ሺܠሻൌܠ

Answer:

Firstly, we will calculate the zero of g(x) then we will substitute the zero of g(x) in p(x) to see if g(x) is a factor of p(x) or not. ሺሻൌͲ ൌ൐൅ʹൌͲ ൌ൐ൌെʹ

Substituting the zero of g(x) in p(x).

ሺെʹሻൌሺെʹሻଷ൅͵ሺെʹሻଶ൅͵ሺെʹሻ൅ͳ

ൌെͺ൅ͳʹെ͸൅ͳ ൌെͳ്Ͳ Therefore, according to the factor theorem g(x) is not a factor of p(x).

(iii) ܘሺܠሻൌܠ૜Ȃ૝ܠ૛൅ܠ൅૟ǡ܏ሺܠሻൌܠ

Answer:

Firstly, we will calculate the zero of g(x) then we will substitute the zero of g(x) in p(x) to see if g(x) is a factor of p(x) or not. ሺሻൌͲ ൌ൐െ͵ൌͲ ൌ൐ൌ͵

Substituting the zero of g(x) in p(x).

ሺ͵ሻൌሺ͵ሻଷെͶሺ͵ሻଶ൅ሺ͵ሻ൅͸

ൌʹ͹െ͵͸൅͵൅͸ ൌͲ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials Therefore, according to the factor theorem g(x) is a factor of p(x). Q3. Find the value of k, if xʹ1 is a factor of p(x) in each of the following cases: (i) ܘሺܠሻൌܠ૛൅ܠ൅ܓ

Answer:

If x-1 is a p(x) factor, then p(1) = 0.

According to Factor Theorem:

ሺͳሻൌሺͳሻଶ൅ሺͳሻ൅ൌͲ ֜ ֜ ֜ (ii) ܘሺܠሻൌ૛ܠ૛൅ܠܓ

Answer:

If x-1 is a p(x) factor, then p(1) = 0.

According to Factor Theorem:

ሺͳሻൌʹሺͳሻଶ൅ሺͳሻ൅ξʹൌͲ ֜ ֜ (iii) ሺሻൌଶȂξʹ൅ͳ

Answer:

If x-1 is a p(x) factor, then p(1) = 0.

According to Factor Theorem:

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ሺͳሻൌሺͳሻଶെξʹሺͳሻ൅ͳൌͲ ֜ (iv) ܘሺܠሻൌܠܓ૛Ȃ૜ܠ൅ܓ

Answer: If x-1 is a p(x) factor, then p(1) = 0.

According to Factor Theorem:

ሺͳሻൌሺͳሻଶȂ͵ሺͳሻ൅ൌͲ ֜ ֜ ֜

Q4. Factorize:

(i) ૚૛ܠ૛Ȃૠܠ

Answer:

Using the procedure of breaking the middle term, we must discover a number with a total of -7 and a product of 12. The integers -3 and -4 are obtained [-3+-4=-7 and -3×-4 = 12]. ͳʹଶȂ͹൅ͳൌͳʹଶെͶെ͵൅ͳ ൌͶሺ͵െͳሻെͳሺ͵െͳሻ ൌሺͶെͳሻሺ͵െͳሻ (ii) ૛ܠ૛൅ૠܠ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

Answer:

Using the procedure of breaking the middle term, we must discover a number with a total of 7 and a product of 6. The integers 6 and 1 are obtained [6+1=7 and 6×1 = 6]. ʹଶ൅͹൅͵ൌʹଶ൅͸൅ͳ൅͵ ൌʹሺ൅͵ሻ൅ͳሺ൅͵ሻ ൌሺʹ൅ͳሻሺ൅͵ሻ (iii) ૟ܠ૛൅૞ܠ

Answer:

Using the procedure of breaking the middle term, we must discover a number with a total of 5 and a product of -36. The integers -4 and -9 are obtained [-4+9 = 5 and -4×9 = -36]. ͸ଶ൅ͷെ͸ൌ͸ଶ൅ͻȂͶȂ͸ ൌ͵ሺʹ൅͵ሻȂʹሺʹ൅͵ሻ ൌሺʹ൅͵ሻሺ͵Ȃʹሻ (iv) ૜ܠ૛Ȃܠ

Answer:

Using the procedure of breaking the middle term, we must discover a number with a total of -1 and a product of -12. The integers -4 and 3 are obtained [-4+3 = -1 and -4×3 = -12]. ͵ଶȂȂͶൌ͵ଶȂͶ൅͵ȂͶ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ൌሺ͵ȂͶሻ൅ͳሺ͵ȂͶሻ ൌሺ͵ȂͶሻሺ൅ͳሻ

Q5. Factorize:

(i) ܠ૜െ૛ܠ૛െܠ

Answer:

Let us assume a function ሺሻൌଷെʹଶെ൅ʹ Now, the factors of 2 are: േͳേʹ ሺሻൌଷȂʹଶȂ൅ʹ

ሺെͳሻൌሺെͳሻଷȂʹሺെͳሻଶȂሺെͳሻ൅ʹ

ൌെͳെʹ൅ͳ൅ʹ ൌͲ So, (x+1) can be said as a factor of the function p(x).

The factorized polynomial will be:

ሺ൅ͳሻሺଶȂ͵൅ʹሻ ൌሺ൅ͳሻሺଶȂȂʹ൅ʹሻ ൌሺ൅ͳሻሺሺെͳሻെʹሺെͳሻሻ ൌሺ൅ͳሻሺെͳሻሺെʹሻ (ii) ܠ૜െ૜ܠ૛െૢܠ

Answer:

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials Let us assume a function ሺሻൌଷെ͵ଶെͻെͷ Now, the factors of 5 are: േͳേͷ .

Using the trial method, we can find that p(5)=0.

Therefore, െͷ is a zero of p(x).

ሺሻൌଷȂ͵ଶȂͻȂͷ

ሺͷሻൌሺͷሻଷȂ͵ሺͷሻଶȂͻሺͷሻȂͷ

ൌͳʹͷെ͹ͷെͶͷെͷ ൌͲ

The factorized polynomial will be:

ሺെͷሻሺଶ൅ʹ൅ͳሻ ൌሺെͷሻሺଶ൅൅൅ͳሻ ൌሺെͷሻሺሺ൅ͳሻ൅ͳሺ൅ͳሻሻ ൌሺെͷሻሺ൅ͳሻሺ൅ͳሻ (iii) ܠ૜൅૚૜ܠ૛൅૜૛ܠ

Answer:

Let us assume a function ሺሻൌଷ൅ͳ͵ଶ൅͵ʹ൅ʹͲ

Now, the factors of 20 are: േͳǡേʹǡേͶǡേͷǡേͳͲേʹͲ .

Using the trial method, we can find that p(-1)=0

Therefore, ൅ͳ is a zero of p(x).

ሺሻൌଷ൅ͳ͵ଶ൅͵ʹ൅ʹͲ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

ሺെͳሻൌሺെͳሻଷ൅ͳ͵ሺെͳሻଶ൅͵ʹሺെͳሻ൅ʹͲ

ൌെͳ൅ͳ͵െ͵ʹ൅ʹͲ ൌͲ

The factorized polynomial will be:

ሺ൅ͳሻሺଶ൅ͳʹ൅ʹͲሻ ൌሺ൅ͳሻሺʹ൅ʹ൅ͳͲ൅ʹͲሻ ൌሺ൅ͳሻሺ൅ʹሻ൅ͳͲሺ൅ʹሻ ൌሺ൅ͳሻሺ൅ʹሻሺ൅ͳͲሻ (iv) ૛ܡ૜൅ܡ૛െ૛ܡ

Answer:

Let us assume a function ሺሻൌʹଷ൅ଶെʹെͳ

Žǁ͕ƚŚĞĨĂĐƚŽƌƐŽĨϮп;оϭͿс-2 are ±1 and ±2.

Using the trial method, we can find that p(1)=0.

Therefore, െͳ is a zero of p(y)

ሺሻൌʹଷ൅ଶȂʹȂͳ

ሺͳሻൌʹሺͳሻଷ൅ሺͳሻଶȂʹሺͳሻȂͳ

ൌʹ൅ͳെʹ ൌͲ

The factorized polynomial will be:

ሺെͳሻሺʹଶ൅͵൅ͳሻ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ൌሺെͳሻሺʹଶ൅ʹ൅൅ͳሻ ൌሺെͳሻሺʹሺ൅ͳሻ൅ͳሺ൅ͳሻሻ ൌሺെͳሻሺʹ൅ͳሻሺ൅ͳሻ

Exercise 2.5:

Q1. Use suitable identities to find the following products: (i) ሺܠ൅૝ሻሺܠ

Answer:

We will be using the identity: ሺ൅ሻሺ൅ሻൌଶ൅ሺ൅ሻ൅

Now, we will use: ൌͶൌͳͲ :

ሺ൅Ͷሻሺ൅ͳͲሻ ൌଶ൅ሺͶ൅ͳͲሻ൅ሺͶൈͳͲሻ ൌଶ൅ͳͶ൅ͶͲ (ii) ሺܠ൅ૡሻሺܠ

Answer:

We will be using the identity: ሺ൅ሻሺ൅ሻൌଶ൅ሺ൅ሻ൅

Now, we will use: ൌͺൌെͳͲ : ሺ൅ͺሻሺെͳͲሻ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

ൌଶ൅ሺͺ൅ሺെͳͲሻሻ൅ሺͺൈሺെͳͲሻሻ

ൌଶ൅ሺͺെͳͲሻȂͺͲ ൌଶെʹെͺͲ (iii) ሺ૜ܠ൅૝ሻሺ૜ܠ

Answer:

We will be using the identity: ሺ൅ሻሺ൅ሻൌଶ൅ሺ൅ሻ൅

Now, we will use: ൌ͵ǡൌͶൌെͷ : ሺ͵൅Ͷሻሺ͵െͷሻ

ൌሺ͵ሻଶ൅ሾͶ൅ሺെͷሻሿ͵൅Ͷൈሺെͷሻ

ൌͻଶ൅͵ሺͶȂͷሻȂʹͲ ൌͻଶȂ͵ȂʹͲ (iv) ቀܡ ૛ቁቀܡ ૛ቁ

Answer:

We will be using the identity: ሺ൅ሻሺെሻൌଶെଶ . Now, we will use: ൌଶൌ͵Ȁʹ . ቀଶ൅ଷ ଶቁቀଶȂଷ ଶቁ ൌሺଶሻଶȂቀଷ ଶቁ ଶ ൌସȂଽ ସ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials Q2. Evaluate the following products without multiplying directly: (i) ૚૙૜ൈ૚૙ૠ

Answer:

ͳͲ͵ൈͳͲ͹ൌሺͳͲͲ൅͵ሻൈሺͳͲͲ൅͹ሻ

We will be using the identity: ሺ൅ሻሺ൅ሻൌଶ൅ሺ൅ሻ൅

Now, we will use: ൌͳͲͲǡൌ͵ൌ͹ .

ͳͲ͵ൈͳͲ͹ൌሺͳͲͲ൅͵ሻൈሺͳͲͲ൅͹ሻ

ൌሺͳͲͲሻଶ൅ሺ͵൅͹ሻͳͲͲ൅ሺ͵ൈ͹ሻ

ൌͳͲͲͲͲ൅ͳͲͲͲ൅ʹͳ ൌͳͳͲʹͳ (ii) ૢ૞ൈૢ૟

Answer:

ͻͷൈͻ͸ൌሺͳͲͲെͷሻൈሺͳͲͲെͶሻ

Now, we will use: ൌͳͲͲǡൌെͷൌെͶ .

ͻͷൈͻ͸ൌሺͳͲͲെͷሻൈሺͳͲͲെͶሻ

ൌሺͳͲͲሻଶ൅ͳͲͲሺെͷ൅ሺെͶሻሻ൅ሺെͷൈെͶሻ

ൌͳͲͲͲͲെͻͲͲ൅ʹͲ ൌͻͳʹͲ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (iii) ૚૙૝ൈૢ૟

Answer:

ͳͲͶൈͻ͸ൌሺͳͲͲ൅ͶሻൈሺͳͲͲȂͶሻ

We will be using the identity: ሺ൅ሻሺെሻൌଶെଶ . Now, we will use: ൌͳͲͲൌͶ .

ͳͲͶൈͻ͸ൌሺͳͲͲ൅ͶሻൈሺͳͲͲȂͶሻ

ൌሺͳͲͲሻଶȂሺͶሻଶ ൌͳͲͲͲͲȂͳ͸ ൌͻͻͺͶ Q3. Factorize the following using appropriate identities: (i) ૢܠ૛൅૟ܡܠ൅ܡ

Answer: ͻଶ൅͸൅ଶൌሺ͵ሻଶ൅ሺʹൈ͵ൈሻ൅ଶ

We will be using the identity: ଶ൅ʹ൅ଶൌሺ൅ሻଶ

Now, we will use: ൌ͵ൌ .

ͻଶ൅͸൅ଶ ൌሺ͵ሻଶ൅ሺʹൈ͵ൈሻ൅ଶ ൌሺ͵൅ሻଶ ൌሺ͵൅ሻሺ͵൅ሻ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (ii) ૝ܡ૛െ૝ܡ

Answer:

ͶଶെͶ൅ͳൌሺʹሻଶȂሺʹൈʹൈͳሻ൅ͳ

We will be using the identity: ଶെʹ൅ଶൌሺെሻଶ

Now, we will use: ൌʹൌͳ .

ͶଶെͶ൅ͳ ൌሺʹሻଶȂሺʹൈʹൈͳሻ൅ͳʹ ൌሺʹȂͳሻଶ ൌሺʹȂͳሻሺʹȂͳሻ (iii) ܠ૛െܡ ૚૙૙

Answer:

ଶȂ୷మ ଵ଴଴ൌଶȂቀ୷ ଵ଴ቁ ଶ We will be using the identity: ଶെଶൌሺെሻሺ൅ሻ

Now, we will use: ൌൌ୷

ଵ଴ . ଶȂ୷మ ଵ଴଴ ൌଶȂቀ୷ ଵ଴ቁ ଶ ൌቀȂ୷ ଵ଴ቁቀ൅୷ ଵ଴ቁ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials Q4. Expand each of the following, using suitable identities: (i) ሺܠ൅૛ܡ൅૝ܢ

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Now, we will use: ൌǡൌʹൌͶ . ሺ൅ʹ൅Ͷሻଶ

ൌଶ൅ሺʹሻଶ൅ሺͶሻଶ൅ሺʹൈൈʹሻ൅ሺʹൈʹൈͶሻ൅ሺʹൈͶൈሻ

ൌଶ൅Ͷଶ൅ͳ͸ଶ൅Ͷ൅ͳ͸൅ͺ (ii) ሺʹെ൅ሻଶ

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Now, we will use: ൌʹǡൌെൌ . ሺʹെ൅ሻଶ

ൌሺʹሻଶ൅ሺെሻଶ൅ଶ൅ሺʹൈʹൈെሻ൅ሺʹൈെൈሻ൅ሺʹൈൈʹሻ

ൌͶଶ൅ଶ൅ଶȂͶȂʹ൅Ͷ (iii) ሺെ૛ܠ൅૜ܡ൅૛ܢ

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Now, we will use: ൌെʹǡൌ͵ൌʹ . NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ሺെʹ൅͵൅ʹሻଶ

ൌሺെʹሻଶ൅ሺ͵ሻଶ൅ሺʹሻଶ൅ሺʹൈെʹൈ͵ሻ൅ሺʹൈ͵ൈʹሻ൅

ሺʹൈʹൈെʹሻ ൌͶଶ൅ͻଶ൅ͶଶȂͳʹ൅ͳʹȂͺ (iv) ሺ૜܉െૠ܊െ܋

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Now, we will use: ൌ͵ǡൌെ͹ൌെ . ሺ͵Ȃ͹Ȃሻଶ

ൌሺ͵ሻଶ൅ሺȂ͹ሻଶ൅ሺȂሻଶ൅ሺʹൈ͵ൈȂ͹ሻ൅ሺʹൈȂ͹ൈȂሻ൅

ሺʹൈȂൈ͵ሻ ൌͻଶ൅Ͷͻଶ൅ଶȂͶʹ൅ͳͶȂ͸ (v) ሺȂ૛ܠ൅૞ܡȂ૜ܢ

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Now, we will use: ൌെʹǡൌͷൌെ͵ . ሺȂʹ൅ͷȂ͵ሻଶ

ൌሺȂʹሻଶ൅ሺͷሻଶ൅ሺȂ͵ሻଶ൅ሺʹൈȂʹൈͷሻ൅ሺʹൈͷൈȂ͵ሻ൅

ሺʹൈȂ͵ൈȂʹሻ ൌͶଶ൅ʹͷଶ൅ͻଶȂʹͲȂ͵Ͳ൅ͳʹ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (vi) ቂ૚ ૝܉ ૛܊ ૛

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Now, we will use: ൌଵ

ସǡൌെଵ ଶൌͳ . ቂଵ ସെଵ ଶ൅ͳቃ ଶ ൌቀଵ ସቁ ଶ൅ቀെଵ ଶቁ ଶ൅ሺͳሻଶ൅ቀʹൈଵ ସൈଵ ଶቁ൅ቀʹൈെଵ ଶൈͳቁ൅ቀʹൈͳൈଵ ସቁ ൌଵ ଵ଺ଶ൅ଵ ସଶ൅ͳଶെଶ ଼െଶ ଶ൅ଶ ସ ൌଵ ଵ଺ଶ൅ଵ ସଶ൅ͳെଵ ସെ൅ଵ ଶ

Q5. Factorize:

(i) ૝ܠ૛൅ૢܡ૛൅૚૟ܢ૛൅૚૛ܡܠȂ૛૝ܢܡȂ૚૟ܢܠ

Answer:

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ

Ͷଶ൅ͻଶ൅ͳ͸ଶ൅ͳʹȂʹͶȂͳ͸

ൌሺʹሻଶ൅ሺ͵ሻଶ൅ሺെͶሻଶ൅ሺʹൈʹൈ͵ሻ൅ሺʹൈ͵ൈെͶሻ൅

ሺʹൈെͶൈʹሻ ൌሺʹ൅͵ȂͶሻଶ ൌሺʹ൅͵ȂͶሻሺʹ൅͵ȂͶሻ

(ii) ૛ܠ૛൅ܡ૛൅ૡܢ૛Ȃ૛ξ૛ܡܠ൅૝ξ૛ܢܡȂૡܢܠ

Answer:

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

We will be using the identity: ሺ൅൅ሻଶൌଶ൅ଶ൅ଶ൅ʹ൅ʹ൅ʹ .

ʹଶ൅ଶ൅ͺଶȂʹξʹ൅ͶξʹȂͺ ൌ൫െξʹ൯ ଶ൅ሺሻଶ൅൫ʹξʹ൯

ଶ൅ሺʹൈെξʹൈሻ൅ሺʹൈൈʹξʹሻ൅

ሺʹൈʹξʹൈെξʹሻ ൌ൫െξʹ൅൅ʹξʹ൯ ଶ

ൌሺെξʹ൅൅ʹξʹሻሺെξʹ൅൅ʹξʹሻ

Q6. Write the following cubes in expanded form:

(i) ሺ૛ܠ

Answer:

We will be using the identity: ሺ൅ሻଷൌଷ൅ଷ൅͵ሺ൅ሻ

ሺʹ൅ͳሻଷ

ൌሺʹሻଷ൅ͳ͵൅ሺ͵ൈʹൈͳሻሺʹ൅ͳሻ

ൌͺଷ൅ͳ൅͸ሺʹ൅ͳሻ ൌͺଷ൅ͳʹଶ൅͸൅ͳ (ii) ሺ૛܉െ૜܊

Answer:

We will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ

ሺʹെ͵ሻଷ

ൌሺʹሻଷെሺ͵ሻଷȂሺ͵ൈʹൈ͵ሻሺʹȂ͵ሻ

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ൌͺଷȂʹ͹ଷȂͳͺሺʹȂ͵ሻ ൌͺଷȂʹ͹ଷȂ͵͸ଶ൅ͷͶଶ (iii) ൬ቀ૜ ૛ቁܠ ૜

Answer:

We will be using the identity: ሺ൅ሻଷൌଷ൅ଷ൅͵ሺ൅ሻ

൬ቀଷ ଶቁ൅ͳ൰ ଷ ൌ൬ቀଷ ଶቁ൰ ଷ ൅ͳ͵൅ቀ͵ൈቀଷ ଶቁൈͳቁ൬ቀଷ ଶቁ൅ͳ൰ ൌଶ଻ ଼ଷ൅ͳ൅ଽ ଶቀଷ ଶ൅ͳቁ ൌଶ଻ ଼ଷ൅ͳ൅ଶ଻ ସଶ൅ଽ ଶ ൌଶ଻ ଼ଷ൅ଶ଻ ସଶ൅ଽ ଶ൅ͳ (iv) ቀܠ ૜ቁܡ ૜

Answer:

We will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ

ቀെଶ ଷቁ ଷ ൌሺሻଷെቀଶ ଷቁ ଷെቀ͵ൈൈଶ ଷቁቀെଶ ଷቁ ൌሺሻଷെ଼ ଶ଻ଷെʹቀെଶ ଷቁ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ൌሺሻଷെ଼ ଶ଻ଷെʹଶ൅ସ ଷଶ Q7. Evaluate the following using suitable identities: (i) ሺૢૢሻ૜

Answer:

The number 99 can be written as 100-1.

Now, we will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ

ሺͻͻሻଷ ൌሺͳͲͲȂͳሻଷ

ൌሺͳͲͲሻଷȂͳ͵Ȃሺ͵ൈͳͲͲൈͳሻሺͳͲͲȂͳሻ

ൌͳͲͲͲͲͲͲȂͳȂ͵ͲͲሺͳͲͲȂͳሻ ൌͳͲͲͲͲͲͲȂͳȂ͵ͲͲͲͲ൅͵ͲͲ ൌͻ͹Ͳʹͻͻ (ii) ሺ૚૙૛ሻ૜

Answer:

The number 102 can be written as 100+2.

Now, we will be using the identity: ሺ൅ሻଷൌଷ൅ଷ൅͵ሺ൅ሻ

ሺͳͲͲ൅ʹሻଷ

ൌሺͳͲͲሻଷ൅ʹଷ൅ሺ͵ൈͳͲͲൈʹሻሺͳͲͲ൅ʹሻ

ൌͳͲͲͲͲͲͲ൅ͺ൅͸ͲͲሺͳͲͲ൅ʹሻ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ൌͳͲͲͲͲͲͲ൅ͺ൅͸ͲͲͲͲ൅ͳʹͲͲ ൌͳͲ͸ͳʹͲͺ (iii) ሺૢૢૡሻ૜

Answer:

The number 998 can be written as 1000-2.

Now, we will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ .

ൌሺͳͲͲͲሻଷȂʹଷȂሺ͵ൈͳͲͲͲൈʹሻሺͳͲͲͲȂʹሻ

ൌͳͲͲͲͲͲͲͲͲͲȂͺȂ͸ͲͲͲሺͳͲͲͲȂʹሻ

ൌͳͲͲͲͲͲͲͲͲͲȂͺെ͸ͲͲͲͲͲͲ൅ͳʹͲͲͲ

ൌͻͻͶͲͳͳͻͻʹ

Q8. Factorize each of the following:

(i) ૡ܉૜൅܊૜൅૚૛܉૛܊൅૟܊܉

Answer:

ͺଷ൅ଷ൅ͳʹଶ൅͸ଶ may be written as ሺʹሻଷ൅ଷ൅͵ሺʹሻଶ൅

͵ሺʹሻሺሻଶ

Now, we will be using the identity: ሺ൅ሻଷൌଷ൅ଷ൅͵ሺ൅ሻ

ͺଷ൅ଷ൅ͳʹଶ൅͸ଶ

ൌሺʹሻଷ൅ଷ൅͵ሺʹሻଶ൅͵ሺʹሻሺሻଶ

ൌሺʹ൅ሻଷ ൌሺʹ൅ሻሺʹ൅ሻሺʹ൅ሻ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (ii) ૡ܉૜Ȃ܊૜Ȃ૚૛܉૛܊൅૟܊܉

Answer:

ͺଷȂଷȂͳʹଶ൅͸ଶ may be written as: ሺʹሻଷȂଷȂ͵ሺʹሻଶ൅͵ሺʹሻሺሻଶ

Now, we will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ .

ͺଷȂଷെͳʹଶ൅͸ଶ

ൌሺʹሻଷȂଷȂ͵ሺʹሻଶ൅͵ሺʹሻሺሻଶ

ൌሺʹȂሻଷ ൌሺʹȂሻሺʹȂሻሺʹȂሻ

(iii) ૛ૠȂ૚૛૞܉૜Ȃ૚૜૞܉൅૛૛૞܉

Answer:

ʹ͹ȂͳʹͷଷȂͳ͵ͷ൅ʹʹͷଶ may be written as ͵ଷȂሺͷሻଷȂ͵ሺ͵ሻଶሺͷሻ൅

͵ሺ͵ሻሺͷሻଶ

Now, we will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ .

ʹ͹ȂͳʹͷଷȂͳ͵ͷ൅ʹʹͷଶ

ൌ͵ଷȂሺͷሻଷȂ͵ሺ͵ሻଶሺͷሻ൅͵ሺ͵ሻሺͷሻଶ

ൌሺ͵Ȃͷሻଷ ൌሺ͵Ȃͷሻሺ͵Ȃͷሻሺ͵Ȃͷሻ

(iv) ૟૝܉૜Ȃ૛ૠ܊૜Ȃ૚૝૝܉૛܊൅૚૙ૡ܊܉

Answer:

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

͸ͶଷȂʹ͹ଷȂͳͶͶଶ൅ͳͲͺଶ may be written as

ሺͶሻଷȂሺ͵ሻଷȂ͵ሺͶሻଶሺ͵ሻ൅͵ሺͶሻሺ͵ሻଶ .

Now, we will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ .

͸ͶଷȂʹ͹ଷȂͳͶͶଶ൅ͳͲͺଶ

ൌሺͶሻଷȂሺ͵ሻଷȂ͵ሺͶሻଶሺ͵ሻ൅͵ሺͶሻሺ͵ሻଶ

ൌሺͶȂ͵ሻଷ ൌሺͶȂ͵ሻሺͶȂ͵ሻሺͶȂ͵ሻ (v) ૠܘ ૛૚૟ቁെቀૢ ૛ቁܘ ૝ቁܘ

Answer:

͹ଷȂቀଵ ଶଵ଺ቁെቀଽ ଶቁଶ൅ቀଵ ସቁ may be written as ሺ͵ሻଷȂቀଵ ଺ቁ ଷȂ͵ሺ͵ሻଶቀଵ ଺ቁ൅

͵ሺ͵ሻቀଵ

଺ቁ ଶ ʹ͹ଷȂቀଵ ଶଵ଺ቁെቀଽ ଶቁଶ൅ቀଵ ସቁ ൌሺ͵ሻଷȂቀଵ ଺ቁ ଷȂ͵ሺ͵ሻଶቀଵ ଺ቁ൅͵ሺ͵ሻቀଵ ଺ቁ ଶ ൌሺ͵Ȃͳ͸ሻଷ ൌሺ͵Ȃͳ͸ሻሺ͵Ȃͳ͸ሻሺ͵Ȃͳ͸ሻ

Q9. Verify:

(i) ܠ૜൅ܡ૜ൌሺܠ൅ܡሻሺܠ૛Ȃܡܠ൅ܡ

Answer:

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

We will be using the identity: ሺ൅ሻଷൌଷ൅ଷ൅͵ሺ൅ሻ

֜ ֜

We will take (x+y) as common:

֜ ֜ (ii) ܠ૜Ȃܡ૜ൌሺܠȂܡሻሺܠ૛൅ܡܠ൅ܡ

Answer:

we will be using the identity: ሺെሻଷൌଷെଷെ͵ሺെሻ .

֜ ֜

We will take (x+y) as common:

֜ ֜

Q10. Factorize each of the following:

(i) ૛ૠܡ૜൅૚૛૞ܢ

Answer:

ʹ͹ଷ൅ͳʹͷଷ may be written as ሺ͵ሻଷ൅ሺͷሻଷ

We will be using the identity ଷ൅ଷൌሺ൅ሻሺଶെ൅ଶሻ

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ʹ͹ଷ൅ͳʹͷଷ ൌሺ͵ሻଷ൅ሺͷሻଷ

ൌሺ͵൅ͷሻሾሺ͵ሻଶȂሺ͵ሻሺͷሻ൅ሺͷሻଶሿ

ൌሺ͵൅ͷሻሺͻଶȂͳͷ൅ʹͷଶሻ (ii) ૟૝ܕ૜Ȃ૜૝૜ܖ

Answer:

͸ͶଷȂ͵Ͷ͵ଷ may be written as ሺͶሻଷെሺ͹ሻଷ .

We will be using the identity ଷെଷൌሺെሻሺଶ൅൅ଶሻ

͸ͶଷȂ͵Ͷ͵ଷ ൌሺͶሻଷȂሺ͹ሻଷ

ൌሺͶെ͹ሻሾሺͶሻଶ൅ሺͶሻሺ͹ሻ൅ሺ͹ሻଶሿ

ൌሺͶെ͹ሻሺͳ͸ଶ൅ʹͺ൅Ͷͻଶሻ Q11. Factorize: ૛ૠܠ૜൅ܡ૜൅ܢ૜Ȃૢܢܡܠ

Answer:

ʹ͹ଷ൅ଷ൅ଷȂͻ may be written as ሺ͵ሻଷ൅ଷ൅ଷȂ͵ሺ͵ሻሺሻሺሻ

We will be using the identity: ଷ൅ଷ൅ଷെ͵ൌሺ൅൅ሻሺଶ൅ଶ൅

ଶെെെሻ ʹ͹ଷ൅ଷ൅ଷȂͻ

ൌሺ͵ሻଷ൅ଷ൅ଷȂ͵ሺ͵ሻሺሻሺሻ

ൌሺ͵൅൅ሻሾሺ͵ሻଶ൅ଶ൅ଶȂ͵ȂȂ͵ሿ

NCERT Solutions for Class 9 Maths Chapter 2- Polynomials

ൌሺ͵൅൅ሻሺͻଶ൅ଶ൅ଶȂ͵ȂȂ͵ሻ

Q12. Verify that: ܠ૜൅ܡ૜൅ܢ૜Ȃ૜ܢܡܠ

૛ቁሺܠ൅ܡ൅ܢሻሾሺܠȂܡሻ૛൅ሺܡȂܢ

ሺܢȂܠ

Answer:

We will be using the identity: ଷ൅ଷ൅ଷെ͵ൌሺ൅൅ሻሺଶ൅ଶ൅

ଶെെെሻ ֜ ൌቀଵ

ଶቁሺ൅൅ሻሺʹଶ൅ʹଶ൅ʹଶȂʹȂʹȂʹሻ

ൌቀଵ

ଶቁሺ൅൅ሻሾሺଶ൅ଶെʹሻ൅ሺଶ൅ଶȂʹሻ൅ሺଶ൅ଶȂʹሻሿ

ൌቀଵ

ଶቁሺ൅൅ሻሾሺȂሻଶ൅ሺȂሻଶ൅ሺȂሻଶሿ

Q13. If ܠ൅ܡ൅ܢൌ૙ , show that ܠ૜൅ܡ૜൅ܢ૜ൌ૜ܢܡܠ

Answer:

We will be using the identity: ଷ൅ଷ൅ଷെ͵ൌሺ൅൅ሻሺଶ൅ଶ൅

ଶെെെሻ As mentioned in the question: ሺ൅൅ሻൌͲ then, ଷ൅ଷ൅ଷെ͵ ൌሺͲሻሺଶ൅ଶ൅ଶȂȂȂሻ ֜ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials ֜ Q14. Without actually calculating the cubes, find the value of each of the following:

(i) ሺെ૚૛ሻ૜൅ሺૠሻ૜൅ሺ૞ሻ૜

Answer:

We already know: if ൅൅ൌͲ then ଷ൅ଷ൅ଷൌ͵ now, in this case: െͳʹ൅͹൅ͷൌͲ If we consider, ൌെͳʹǡൌ͹ൌͷ then:

ሺെͳʹሻଷ൅ሺ͹ሻଷ൅ሺͷሻଷൌ͵

ൌ͵ൈെͳʹൈ͹ൈͷ ൌെͳʹ͸Ͳ

(ii) ሺ૛ૡሻ૜൅ሺെ૚૞ሻ૜൅ሺെ૚૜ሻ૜

Answer:

We already know: if ൅൅ൌͲ then ଷ൅ଷ൅ଷൌ͵ now, in this case: ʹͺെͳͷെͳ͵ൌͲ

ሺʹͺሻଷ൅ሺെͳͷሻଷ൅ሺെͳ͵ሻଷൌ͵

ൌͲ൅͵ሺʹͺሻሺെͳͷሻሺെͳ͵ሻ ൌͳ͸͵ͺͲ Q15. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: NCERT Solutions for Class 9 Maths Chapter 2- Polynomials (i) Area: ૛૞܉૛െ૜૞܉

Answer:

Using the technique of separating the middle term, we must discover a number with a sum of -35 and a product of 25x12=300.

As numbers, we obtain -15 and -20.

ʹͷଶȂ͵ͷ൅ͳʹ ൌʹͷଶȂͳͷെʹͲ൅ͳʹ ൌͷሺͷȂ͵ሻȂͶሺͷȂ͵ሻ ൌሺͷȂͶሻሺͷȂ͵ሻ The expression that might account for the length will be: ሺͷെͶሻ The expression that might account for the width will be: ሺͷെ͵ሻ (ii) Area: ૜૞ܡ૛൅૚૜ܡ

Answer:

Using the technique of separating the middle term, we must discover a number with a sum of 13 and a product of 35x-12=-420.

As numbers, we obtain -15 and 28.

͵ͷଶ൅ͳ͵Ȃͳʹ ൌ͵ͷଶȂͳͷ൅ʹͺȂͳʹ ൌͷሺ͹Ȃ͵ሻ൅Ͷሺ͹Ȃ͵ሻ ൌሺͷ൅Ͷሻሺ͹Ȃ͵ሻ The expression that might account for the length will be: ሺͷ൅Ͷሻ NCERT Solutions for Class 9 Maths Chapter 2- Polynomials The expression that might account for the width will be: ሺ͹െ͵ሻ Q16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume: ૜ܠ૛Ȃ૚૛ܠ Answer: The given expression may be represented as: ͵ሺെͶሻ The expression that might account for the length will be: 3 The expression that might account for the width will be: x The expression that might account for the height will be: ሺെͶሻ (ii) Volume: ૚૛ܡܓ૛൅ૡܡܓȂ૛૙ܓ

Answer: The given expression may be represented as: Ͷሺ͵ଶ൅ʹെͷሻ

Now, we can write the term ሺ͵ଶ൅ʹെͷሻ as: ሺ͵ଶ൅ͷെ͵െͷሻ

ൌͶሺ͵ଶ൅ͷȂ͵Ȃͷሻ ൌͶሾሺ͵൅ͷሻȂͳሺ͵൅ͷሻሿ ൌͶሺ͵൅ͷሻሺȂͳሻ The expression that might account for the length will be: 4k The expression that might account for the width will be: ሺ͵൅ͷሻ The expression that might account for the height will be: ሺെͳሻ