Ncert mathematics solutions class 9 Solutions for Chapter 2 polynomials

Write the coefficients of x² in 2+x²+x .

Classify x–x³ as linear, quadratic and cubic polynomials.

Classify y+y²+4 as linear, quadratic and cubic polynomials.

Classify 3t as linear, quadratic and cubic polynomials.

Classify x²+x as linear, quadratic and cubic polynomials.

Is 4x^2-3x+7 a polynomial in one variable or not ?

Is x^{10}+y^3+t^{50} a polynomial in one variable or not ?

Is 3\sqrt{t}+t\sqrt{2} a polynomial in one variable or not ?

Is y^2+\sqrt{2} a polynomial in one variable or not ?

Is y+\frac 2y a polynomial in one variable or not ?

Find monomial of degree 100.

Find binomial of degree 35.

Write the coefficients of x2 in \sqrt{2}x-1.

Write the coefficients of x² in 2–x²+x³ .

Write the coefficients of x2 in \frac{\pi}{2}x^2+x.

Write the degree of 5t–√7 .

Write the degree of 3.

Write the degree of 4–y² .

Write the degree of 5x³+4x²+7x

Classify 1+x as linear, quadratic and cubic polynomials.

Classify 7x³ as linear, quadratic and cubic polynomials.

Classify r² as linear, quadratic and cubic polynomials.

Which of the following expressions are polynomials in one variable and which are not?

State reasons for your answer.

\text { (i) } 4 x^{2}-3 x+7

\text { (ii) } \quad y^{2}+\sqrt{2}

\text { (iii) } \quad 3 \sqrt{t}+t \sqrt{2}

\text { (iv) } \quad y+\frac{2}{y}

\text { (v) } x^{10}+y^{3}+t^{50}

Write the coefficients of x^2 in each of the following:

\text { (i) } 2+x^{2}+x

\text { (ii) } 2-x^{2}+x^{3}

\text { (iii) } \quad \frac{\pi}{2} x^{2}+x

\text { (iv) } \sqrt{2} x-1

Find p(0), p(1) and p(2) for each of the following polynomials:

\text { (i) } p(y)=y^{2}-y+1

\text { (ii) } \mathrm{p}(\mathrm{t})=2+\mathrm{t}+2 \mathrm{t}^{2}-\mathrm{t}^{3}

\text { (iii)p }(x)=x^{3}

\text { (iv) } p(x)=(x-1)(x+1)

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Write the degree of each of the following polynomials:

\begin{aligned} &\text { (i) }\\ &5 x^{3}+4 x^{2}+7 x \end{aligned}

\text { (ii) } \quad 4-y^{2}

\text { (iii) } 5 t-\sqrt{7}

\text { (iv) } 3

Find p(0) of p(y)=y²−y+1.

Find the value of the polynomial (x)=5x−4x²+3 when x = 0.

Find the value of the polynomial (x)=5x−4x²+3 when x = 2 .

Find the value of the polynomial (x)=5x−4x²+3 when x = – 1 .

Find p(0) of p(t)=2+t+2t²−t³ .

Find p(1) of p(y)=y²−y+1.

Find p(0) of p(x) = (x−1)(x+1)

Find p(1) of p(x) = (x−1)(x+1)

Find p(2) of p(x) = (x−1)(x+1)

Find the zero of the polynomial of p(x) = cx+d, c ≠ 0, c, d are real numbers.

Find p(0) of p(x)=x³ .

Find p(1) of p(x)=x³ .

Find p(2) of p(y)=y²−y+1.

Find p(2) of p(x)=x³ .

Find p(1) of p(t)=2+t+2t²−t³ .

Verify whether p(x)=2x+1,x=\frac{1}{2} are zeroes of the polynomial.

Verify whether p(x)=3x^2-1,x=-\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}} are zeroes of the polynomial.

Verify whether p(x)=x²−1, x=1, −1 are zeroes of the polynomial.

Verify whether p(x)=3x+1, x=−1/3 are zeroes of the polynomial.

Verify whether p(x)=5x–π, x = 4/5 are zeroes of the polynomial.

Verify whether p(x) = (x+1)(x–2), x =−1, 2 are zeroes of the polynomial.

Verify whether p(x) = x², x = 0 are zeroes of the polynomial.

Verify whether p(x)=lx+m,x=-\frac{m}{l} **are zeroes of the polynomial.**

Find the zero of the polynomial of p(x) = 3x.

Find the zero of the polynomial of p(x) = 3x–2.

Find the zero of the polynomial of p(x) = 2x+5.

Find the zero of the polynomial of p(x) = x-5.

Find the zero of the polynomial of p(x) = x+5.

Find the zero of the polynomial of p(x) = ax, a0.

Find p(2) of p(t)=2+t+2t²−t³ .

Find the value of the polynomial (x)=5 x-4 x^{2}+3

(i) x= 0

(ii) x = – 1

(iii) x = 2

Find the remainder when x³−ax²+6x−a is divided by x-a.

Check whether 7+3x is a factor of 3x³+7x.

Find the remainder when x³+3x²+3x+1 is divided by 5+2x.

Find the remainder when x³+3x²+3x+1 is divided by x+1.

Find the remainder when x³+3x²+3x+1 is divided by x−1/2.

Factorize 3x²–x–4

Find the value of k, if x–1 is a factor of p(x) in p(x)=kx²–3x+k.

Factorize 12x²–7x+1

Factorize 6x²+5x-6

Find the value of k, if x–1 is a factor of p(x) in p(x) = x²+x+k.

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in p(x)=x³–4x²+x+6, g(x) = x–3.

State wether x³ – x²– (2+√2)x +√2 is a factor of (x + 1).

State wether x⁴+3x³+3x²+x+1 is a factor of (x + 1).

State wether x⁴+x³+x²+x+1 is a factor of (x + 1).

Find the value of k, if x–1 is a factor of p(x) in p(x) = 2x²+kx+√2.

State wether x³+x²+x+1 is a factor of (x + 1).

Use the Factor Theorem to determine whether g(x) is a factor of p(x)=x³+3x²+3x+1, g(x) = x+2.

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in p(x) = 2x³+x²–2x–1, g(x) = x+1.

Factorize 2x²+7x+3

Factorize x³–3x²–9x–5

Factorize x³–2x²–x+2

Factorize x³+13x²+32x+20

Factorize 2y³+y²–2y–1

Find the value of k, if x–1 is a factor of p(x) in p(x) = kx²–√2x+1.

Find the value of k, if x – 1 is a factor of p(x) in each of the following cases:

\text { (i) } p(x)=x^{2}+x+k

\text { (ii) } p(x)=2 x^{2}+k x+\sqrt{2}

(\text { iii }), \mathbf{p}(\mathbf{x})=\mathbf{k} \mathbf{x}^{2}-\sqrt{2} \mathbf{x}+\mathbf{1}

(\mathrm{iv}) \mathrm{p}(\mathrm{x})=\mathrm{k} \mathrm{x}^{2}-3 \mathrm{x}+\mathrm{k}

Factorize:

\text { (i) } 12 x^{2}-7 x+1

\text { (ii) } 2 x^{2}+7 x+3

(\text { iii }) 6 x^{2}+5 x-6

\text { (iv) } 3 x^{2}-x-4

Factorise 64a³–27b³–144a²b+108ab²

Find the factorization of 9x²+6xy+y².

Evaluate the product without multiplying directly 213 x 189.

Expand using suitable identity: \left(-\frac{x}{5}+\frac{y}{3}-\frac{z}{2}\right)^2

Evaluate: (109)³

Use suitable identities to find \left(\frac{3x}{4}+\frac{4}{7}\right)\left(\frac{3x}{4}-\frac{5}{8}\right)

Find the factorization of x^{ 2}-\frac {y^2}{100}.

\text{Factorize: }\frac{64}{125}a^3-\frac{96}{25}a^2+\frac{48}{5}a-8

\text{Factorize: }8p^3+\frac{12}{5}p^2+\frac{6}{25}p+\frac{1}{125}

Use suitable identities to find (3x+4)(3x–5).

Use suitable identities to find (y²+3/2)(y²-3/2).

Use suitable identities to find (x+4)(x +10).

Use suitable identities to find (x+8)(x –10).

Evaluate 103×107.

Expand (–2x+5y–3z)²

Find the factorization of x^{ 2}-\frac {y^2}{100}.

Find the factorization of 9x²+6xy+y².

Find the factorization of 4y²−4y+1.

Expand (\frac 14 a-\frac 1 2b+1)^2

Expand (3a –7b–c)²

Expand (−2x+3y+2z)²

Expand (x+2y+4z)²

Expand (2x−y+z)²

Factorize 2x²+y²+8z²–2√2xy+4√2yz–8xz.

Factorize 4x²+9y²+16z²+12xy–24yz–16xz.

Evaluate 95×96.

Find the cubic form of (\frac 3 2x+1)^3

Find the cubic form of (2a−3b)³

Verify wether x³+y³ = (x+y)(x²–xy+y²)

Evaluate (998)³

Verify wether x³–y³ = (x–y)(x²+xy+y²)

Factorise 27p^3-\frac{1}{216}-\frac{9}{2}p^2+\frac{1}{4}p.

Factorize 27y³+125z³

Evaluate (102)³

Factorise 8a³+b³+12a²b+6ab²

Give possible expressions for the length and breadth of the following rectangle:

Area : 25a²–35a+12

Evaluate (99)³

Factorise 8a³–b³–12a²b+6ab²

Factorise 27–125a³–135a+225a²

Factorize 64m³–343n³

Factorise: 27x³+y³+z³–9xyz

If x+y+z = 0, check if x³+y³+z³ = 3xyz.

Without actually calculating the cubes, find the value of (28)³+(−15)³+(−13)³

Without actually calculating the cubes, find the value of (−12)³+(7)³+(5)³

Give possible expressions for the length and breadth of the following rectangle:

Area : 35y²+13y–12

Verify that:

x^3+y^3+z^3-3xyz=\frac{1}{2}\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]

What are the possible expressions for the dimensions of the cuboid whose volume is given below?

Volume : 12ky²+8ky–20k

What are the possible expressions for the dimensions of the cuboid whose volume is given below?

Volume : 3x²–12x

Expand (x−\frac 2 3y)^3

Find the cubic form of (2x+1)³

Evaluate 104×107.

Expand each of the following, using suitable identities:

(i)(x+2 y+4 z)^{2}

\text { (ii) }(2 x-y+z)^{2}

(i i i)(-2 x+3 y+2 z)^{2}

(i v)(3 a-7 b-c)^{2}

(v)(-2 x+5 y-3 z)^{2}

(v i)\left(\frac{1}{4} a-\frac{1}{2} b+1\right)^{2}

\text { (i) }(2 x+1)^{3}

\text { (ii) }(2 a-3 b)^{3}

(\mathrm{iii})\left(\frac{3}{2} \mathrm{x}+1\right)^{3}

(\mathrm{iv})\left(\mathrm{x}-\frac{2}{3} \mathrm{y}\right)^{3}

Evaluate the following using suitable identities:

\text { (i) }(99)^{3}

\text { (ii) }(102)^{3}

(\text { iii })(998)^{3}

Factorise each of the following:

\text { (i) } 8 a^{3}+b^{3}+12 a^{2} b+6 a b^{2}

\text { (ii) } 8 a^{3}-b^{3}-12 a^{2} b+6 a b^{2}

(i i i) 27-125 a^{3}-135 a+225 a^{2}

\text { (iv) } 64 a^{3}-27 b^{3}-144 a^{2} b+108 a b^{2}

(v) 27 p^{3}-\frac{1}{216}-\frac{9}{2} p^{2}+\frac{1}{4} p

Verify:

\text { (i) } x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)

(\mathrm{ii}) \mathrm{x}^{3}-\mathrm{y}^{3}=(\mathrm{x}-\mathrm{y})\left(\mathrm{x}^{2}+\mathrm{xy}+\mathrm{y}^{2}\right)

\text { Factorise }: 27 x^{3}+y^{3}+z^{3}-9 x y z

Without actually calculating the cubes, find the value of each of the following:

\text { (i) }(-12)^{3}+(7)^{3}+(5)^{3}

\text { (ii) }(28)^{3}+(-15)^{3}+(-13)^{3}

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

\text { (i) Volume }: 3 x^{2}-12 x

\text { (ii) Volume : } 12 \mathrm{ky}^{2}+8 \mathrm{ky}-20 \mathrm{k}