Balbharati solutions for Algebra (Mathematics 1) [English] 9 Standard Maharashtra State Board chapter 3 - Polynomials [Latest edition]

State whether the given algebraic expression are polynomial? Justify.

`y + 1/y`

State whether the given algebraic expression are polynomial? Justify.

`2 - 5 sqrt x`

State whether the given algebraic expression are polynomial? Justify.

`x^2 + 7x + 9`

State whether the given algebraic expression is polynomial. Justify.

`2m^(-2) + 7m - 5`

State whether the given algebraic expression is polynomial? Justify.

10

Write the coefficient of m³ in the given polynomial.

`m^3`

Write the coefficient of m³ in the given polynomial.

`(-3)/2+m - sqrt 3 m^3`

Write the coefficient of m³ in the given polynomial.

`-2/3 m^3 - 5 m^2 + 7m -1`

Write the polynomial in x using the given information.

Monomial with degree 7

Write the polynomial in x using the given information.

Binomial with degree 35

Write the polynomial in x using the given information.

Trinomial with degree 8

Write the degree of the given polynomial.

`sqrt 5`

Write the degree of the given polynomial.

x°

Write the degree of the given polynomial.

`x^2`

Write the degree of the given polynomial.

`sqrt2m^10 - 7`

Write the degree of the given polynomial.

`2p - sqrt 7`

Write the degree of the given polynomial.

`7y - y^3 + y^5`

Write the degree of the given polynomial.

xyz + xy - z

Write the degree of the given polynomial.

`m^3n^7 -3m^5n + mn`

Classify the following polynomial as linear, quadratic and cubic polynomial.

2x² + 3x + 1

Classify the following polynomial as linear, quadratic and cubic polynomial.

5p

Classify the following polynomial as linear, quadratic and cubic polynomial.

`sqrt 2  y - 1/2`

Classify the following polynomial as linear, quadratic and cubic polynomial.

`m^3 + 7m^2 + 5/2m - sqrt 7`

Classify the Following Polynomial as Linear, Quadratic and Cubic Polynomial.

`a^2`

Classify the following polynomial as linear, quadratic and cubic polynomial.

`3r^3`

Write the following polynomial in standard form.

 m³ + 3 + 5m

Write the following polynomial in standard form.

`-7y + y^5 + 3y^3 - 1/2 +2y^4 -y^2`

Write the following polynomial in coefficient form.

`x^3 - 2`

Write the following polynomial in coefficient form.

5y

Write the following polynomial in coefficient form.

2m⁴ - 3m² + 7

Write the following polynomial in coefficient form.

`-2/3`

Write the polynomial in standard form.

(1, 2, 3)

Write the polynomial in standard form.

(5, 0, 0, 0, -1) 

Write the polynomial in standard form.

(-2, 2, -2, 2)

Write the appropriate polynomials in the boxes.

There are ‘a’ trees in the village Lat. If the number of trees increases every year by ‘b’, then how many trees will there be after ‘x’ years?

The tens and units place of a two digit number is m and n respectively. Write the polynomial which represents the two digit number.

Add the given polynomials.

`x^3 - 2x^2 - 9 ;  5x^3 + 2x + 9`

Add the given polynomial.

`-7m^4 +5m^3 + sqrt2 ;   5m^4 - 3m^3 + 2m^2 + 3m - 6`

Add the given polynomial.

`2y^2 + 7y + 5  ;  3y + 9  ;  3y^2 - 4y - 3`

Subtract the second polynomial from the first.

`x^2 - 9x + sqrt 3  ;  -19x + sqrt 3 +7x^2`

Subtract the second polynomial from the first.

`2ab^2 + 3a^2b - 4ab ; 3ab - 8ab^2 + 2a^2b`

Multiply the given polynomial.

`2x  ;  x^2 - 2x - 1`

Multiply the given polynomial.

`x^5 - 1  ;  x^3 +2x^2 + 2`

Multiply the given polynomial.

`2y + 1 ; y^2 - 2y^3 + 3y`

Divide first polynomial by second polynomial and write the answer in the form ‘Dividend = Divisor × Quotient + Remainder’.

`x^3 - 64 ; x - 4`

Divide first polynomial by second polynomial and write the answer in the form ‘Dividend = Divisor × Quotient + Remainder’.

`5x^5 + 4x^4 - 3x^3 + 2x^2 +2;x^2-x`

There is a rectangular farm with length `(2a^2 + 3b^2)` metre and breadth `(a^2 + b^2)` metre. The farmer used a square shaped plot of the farm to build a house. The side of the plot was `(a^2 -  b^2)` metre.
What is the area of the remaining part of the farm?

Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(2m^2 - 3m + 10) ÷ (m - 5)`

Divide of the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(x^4 + 2x^3 +3x^2 + 4x + 5) ÷ (x + 2)`

Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(y^3 - 216) ÷ (y - 6)`

Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(2x^4 + 3x^3 + 4x - 2x^2) ÷ (x + 3)`

Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(x^4 - 3x^2 - 8) ÷ (x + 4)`

Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(y^3 - 3y^2 + 5y - 1) ÷ (y - 1)`

For x = 0 find the value of the polynomial x² - 5x + 5.

If `p(y) =y^2 - 3sqrt2y + 1` then find `p (3sqrt 2)`.

If `p(m) = m^3 + 2m^2 - m + 10` then `p(a) + p(-a) =` ?

If p(y) = 2y³ − 6y² − 5y + 7 then find p(2).

Find the value of the polynomial 2x − 2x³ + 7 using given values for x.

x = 3

Find the value of the polynomial 2x - 2x³ + 7 using given values for x.

x = -1

Find the value of the polynomial 2x − 2x³ + 7 using given values for x.

x = 0

For the following polynomial, find p(1), p(0) and p(-2).

`p(x) = x^3`

For the following polynomial, find p(1), p(0) and p(-2).

p(y) = y² − 2y + 5

For the following polynomial, find p(1), p(0) and p(-2).

p(x) = x⁴ − 2x² − x

If the value of the polynomial m³ + 2m + a is 12 for m = 2,  then find the value of a.

For the polynomial mx² − 2x + 3 if p(−1) = 7 then find m.

Divide the first polynomial by the second polynomial and find the remainder using remainder theorem.

(x² − 7x + 9) ; (x + 1)

Divide the first polynomial by the second polynomial and find the remainder using remainder theorem.

(2x³ − 2x² + ax − a) ; (x − a)

Divide the first polynomial by the second polynomial and find the remainder using remainder theorem.

(54m³ + 18m² − 27m + 5) ; (m − 3)

If the polynomial y³ − 5y² + 7y + m is divided by y + 2 and the remainder is 50 then find the value of m.

Use factor theorem to determine whether x + 3 is factor of x ² + 2x − 3 or not. 

If (x - 2) is a factor of x³ − mx² + 10x − 20 then find the value of m.

By using factor theorem in the following example, determine whether q(x) is a factor p(x) or not.

p(x) = x³ − x² − x − 1, q(x) = x − 1

By using factor theorem in the following example, determine whether q(x) is a factor p(x) or not.

p(x) = 2x³ − x² − 45, q(x) = x − 3

If ( x³¹ + 31) is divided by (x + 1) then find the remainder. 

Show that m − 1 is a factor of m²¹ − 1 and m²² − 1.

If x - 2 and  `x - 1/2` both are the factors of the polynomial  nx² − 5x + m, then show that  m = n = 2

If p(x) = 2 + 5x then p(2) + p(−2) − p(1).

If `p(x) = 2x^2 - 5sqrt 3 x + 5` then `p(5 sqrt 3)`.

Find the Factors of the Polynomial Given Below.

2x² + x – 1

Find the factor of the polynomial given below.

2m² + 5m – 3

Find the factor of the polynomial given below.

12x² + 61x + 77

Find the factor of the polynomial given below.

3y² – 2y – 1

Find the factor of the polynomial given below.

`sqrt 3 x^2 + 4x + sqrt 3`

Find the factor of the polynomial given below.

`1/2x^2 - 3x + 4`

Factorize the following polynomial.

(x² – x)² – 8 (x² – x) + 12

Factorize the following polynomial.

(x – 5)² – (5x – 25) – 24

Factorize the following polynomial.

 (x² – 6x)² – 8 (x² – 6x + 8) – 64

Factorize the following polynomial.

(x² – 2x + 3) (x² – 2x + 5) – 35 

Factorize the following polynomial.

(y + 2) (y – 3) (y + 8) (y + 3) + 56 

Factorize the following polynomial.

 (y² + 5y) (y² + 5y – 2) – 24

Factorize the following polynomial.

(x – 3) (x – 4)² (x – 5) – 6 

Which of the following is a polynomial?

What is the degree of the polynomial `sqrt 7`?

What is the degree of the 0 polynomial?

What is the degree of the polynomial 2x² + 5x³ + 7? 

What is the coefficient form of `x^3 - 1`?

`p(x) = x^2 - 7 sqrt 7 x + 3  "then " p(7 sqrt 7) = ?`

When x = −1, what is the value of the polynomial 2x³ + 2x? 

If x - 1 is a factor of the polynomial `3x^2 + mx `  then find the value of m. 

Multiply (x² - 3) (2x - 7x ³ + 4) and write the degree of the product. 

Which of the following is a linear polynomial?

Write the degree of the polynomial for the following.

5 + 3x⁴ 

Write the degree of the polynomial for the following.

7

Write the degree of the polynomial for the following.

ax⁷ + bx⁹ (a, b are constants.)

Write the following polynomial in standard form.

4x² + 7x⁴ - x³ - x + 9

Write the following polynomial in standard form. 

p + 2p³ + 10p² + 5p⁴ - 8

Write the following polynomial in coefficient form.

x⁴ + 16  

Write the following polynomial in coefficient form.

m⁵ + 2m² + 3m + 15

Write the index form of the polynomial using variable x from its coefficient form.

(3, -2, 0, 7, 18)

Write the index form of the polynomial using variable x from its coefficient form. 

(6, 1, 0, 7)

Write the index form of the polynomial using variable x from its coefficient form.

(4, 5, -3, 0)

Add the following polynomial.

`7x^4 - 2x^3 + x + 10 ; 3x^4 + 15x^3 + 9x^2 - 8x + 2`

Add the following polynomial.

`3p^3q+ 2p^2q + 7; 2p^2q + 4pq - 2p^3q`

Subtract the second polynomial from the first.

`5x^2 - 2y + 9 ; 3x^2 + 5y - 7`

Subtract the second polynomial from the first.

`2x^2 + 3x + 5 ; x^2 -2x + 3`

Multiply the following polynomial.

`(m^3 - 2m + 3)(m^4 - 2m^2 + 3m + 2)`

Multiply the following polynomial.

`(5m^3 - 2) (m^2 - m + 3)`

Divide polynomial 3x³ - 8x² + x + 7 by x - 3 using synthetic method and write the quotient and remainder.

For which the value of m, x + 3 is the factor of the polynomial x³ − 2mx + 21?

At the end of the year 2016, the population of villages Kovad, Varud, Chikhali is 5x² - 3y², 7y² + 2xy and 9x² + 4xy respectively. At the beginning of the year 2017, x² + xy - y², 5xy and 3x² + xy persons from each of the three villages respectively went to another village for education then what is the remaining total population of these three villages?

Polynomials bx² + x + 5 and bx³ − 2x + 5 are divided by polynomial x - 3 and the remainders are m and n respectively. If m − n = 0 then find the value of b.

Simplify.

(8m² + 3m − 6) − (9m − 7) + (3m² − 2m + 4)

Which polynomial is to be subtracted from x² + 13x + 7 to get the polynomial  3x² + 5x − 4?

Which polynomial is to be added to 4m + 2n + 3 to get the polynomial 6m + 3n + 10?