# Polynomials - Class 9 - Mathematics

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## Exercise 2.1 - Polynomials | NCERT | Mathematics | Class 9

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

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

(ii) $y^{2}+\sqrt{2}$

(iii) $3 \sqrt{t}+t \sqrt{2}$

(iv) $y+\frac{2}{y}$

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

Here's an analysis for each expression to determine if it's a polynomial in one variable or not, alongside reasons:

### (i) $4x^2 - 3x + 7$

This is a polynomial in one variable ($x$) because:

- It is expressed in terms of a single variable, $x$.
- All the powers of $x$ are whole numbers (2 and 1).
- It consists of a finite number of terms, each a product of a constant and a power of $x$.

### (ii) $y^2 + \sqrt{2}$

This is a polynomial in one variable ($y$) because:

- It is expressed in terms of a single variable, $y$.
- The power of $y$ is a whole number (2).
- $\sqrt{2}$ is a constant term, which is allowed in polynomials.

### (iii) $3\sqrt{t} + t\sqrt{2}$

This is not a polynomial for several reasons:

- Although it's in terms of a single variable, $t$, the term $3\sqrt{t}$ involves a square root of the variable, which means the power of $t$ is $\frac{1}{2}$, not a whole number.

### (iv) $y + \frac{2}{y}$

This is not a polynomial because:

- It includes a term with $y$ in the denominator ($\frac{2}{y}$), which implies a negative exponent (-1) for $y$. Polynomials require all powers of the variable to be non-negative whole numbers.

### (v) $x^{10} + y^3 + t^{50}$

This is not a polynomial in one variable because:

- It involves three different variables ($x$, $y$, and $t$), making it a polynomial in three variables, not one.

In summary:

- Expressions (i) and (ii) are polynomials in one variable.
- Expressions (iii) and (iv) are not polynomials due to the presence of non-whole number and negative exponents, respectively.
- Expression (v) is not a polynomial in one variable due to involving more than one variable.

Write the coefficients of $x^{2}$ in each of the following:

(i) $2+x^{2}+x$

(ii) $2-x^{2}+x^{3}$

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

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

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Sign up nowGive one example each of a binomial of degree 35 , and of a monomial of degree 100 .

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Sign up nowWrite the degree of each of the following polynomials:

(i) $5 x^{3}+4 x^{2}+7 x$

(ii) $4-y^{2}$

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

(iv) 3

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Sign up nowClassify the following as linear, quadratic and cubic polynomials:

(i) $x^{2}+x$

(ii) $x-x^{3}$

(iii) $y+y^{2}+4$

(iv) $1+x$

(v) $3 t$

(vi) $r^{2}$

(vii) $7 x^{3}$

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## Exercise 2.2 - Polynomials | NCERT | Mathematics | Class 9

Find the value of the polynomial $5 x-4 x^{2}+3$ at

(i) $x=0$

(ii) $x=-1$

(iii) $x=2$

The value of the polynomial $5x - 4x^2 + 3$ at the given points are:

(i) At $x=0$, the value is $3$.

(ii) At $x=-1$, the value is $-6$.

(iii) At $x=2$, the value is $-3$.

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

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

(ii) $p(t)=2+t+2 t^{2}-t^{3}$

(iii) $p(x)=x^{3}$

(iv) $p(x)=(x-1)(x+1)$

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Verify whether the following are zeroes of the polynomial, indicated against them.

(i) $p(x)=3 x+1, x=-\frac{1}{3}$

(ii) $p(x)=5 x-\pi, x=\frac{4}{5}$

(iii) $p(x)=x^{2}-1, x=1,-1$

(iv) $p(x)=(x+1)(x-2), x=-1,2$

(v) $p(x)=x^{2}, x=0$

(vi) $p(x)=l x+m, x=-\frac{m}{l}$

(vii) $p(x)=3 x^{2}-1, x=-\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}$

(viii) $p(x)=2 x+1, x=\frac{1}{2}$

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Find the zero of the polynomial in each of the following cases:

(i) $p(x)=x+5$

(ii) $p(x)=x-5$

(iii) $p(x)=2 x+5$

(iv) $p(x)=3 x-2$

(v) $p(x)=3 x$

(vi) $p(x)=a x, a \neq 0$

(vii) $p(x)=c x+d, c \neq 0, c, d$ are real numbers.

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## Exercise 2.3 - Polynomials | NCERT | Mathematics | Class 9

Determine which of the following polynomials has $(x+1)$ a factor :

(i) $x^{3}+x^{2}+x+1$

(ii) $x^{4}+x^{3}+x^{2}+x+1$

(iii) $x^{4}+3 x^{3}+3 x^{2}+x+1$

(iv) $x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2}$

For a polynomial to have $(x + 1)$ as a factor, plugging $x = -1$ into the polynomial must result in $0$.

- For the polynomial $x^{3}+x^{2}+x+1$, substituting $x = -1$ does not result in $0$, hence $(x+1)$ is
**not**a factor. - For $x^{4}+x^{3}+x^{2}+x+1$, substituting $x = -1$ results in $0$. Thus, $(x+1)$ is a factor.
- For $x^{4}+3 x^{3}+3 x^{2}+x+1$, substituting $x = -1$ also results in $0$, indicating that $(x+1)$ is a factor.
- Finally, for $x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2}$, substituting $x = -1$ does not yield a result of $0$, which means $(x+1)$ is
**not**a factor.

Therefore, the polynomials which have $(x+1)$ as a factor are:

(ii) $x^{4}+x^{3}+x^{2}+x+1$ and (iii) $x^{4}+3 x^{3}+3 x^{2}+x+1$.

Use the Factor Theorem to determine whether $g(x)$ is a factor of $p(x)$ in each of the following cases:

(i) $p(x)=2 x^{3}+x^{2}-2 x-1, g(x)=x+1$

(ii) $p(x)=x^{3}+3 x^{2}+3 x+1, g(x)=x+2$

(iii) $p(x)=x^{3}-4 x^{2}+x+6, g(x)=x-3$

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

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

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

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

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

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Factorise :

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

(ii) $2 x^{2}+7 x+3$

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

(iv) $3 x^{2}-x-4$

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Factorise :

(i) $x^{3}-2 x^{2}-x+2$

(ii) $x^{3}-3 x^{2}-9 x-5$

(iii) $x^{3}+13 x^{2}+32 x+20$

(iv) $2 y^{3}+y^{2}-2 y-1$

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## Exercise 2.4 - Polynomials | NCERT | Mathematics | Class 9

Use suitable identities to find the following products:

(i) $(x+4)(x+10)$

(ii) $(x+8)(x-10)$

(iii) $(3 x+4)(3 x-5)$

(iv) $\left(y^{2}+\frac{3}{2}\right)\left(y^{2}-\frac{3}{2}\right)$

(v) $(3-2 x)(3+2 x)$

To find the products for each given pair, we can use the algebraic identities:

$$ (a + b)(a + c) = a^2 + a(b + c) + bc $$

$$ (a + b)(a - b) = a^2 - b^2 $$

Let's compute each of the products using these identities:

(i) $(x+4)(x+10)$:

Using the first identity: $$ x^2 + x(4 + 10) + 4\cdot10 = x^2 + 14x + 40 $$

(ii) $(x+8)(x-10)$:

Using the second identity: $$ x^2 - 8\cdot10 = x^2 - 80 $$

(iii) $(3x+4)(3x-5)$:

Using the first identity but notice the coefficients in front of x are the same in both terms, so it's also applicable to treat it like the difference of two squares with an additional middle term. $$ (3x)^2 + 3x(4 - 5) + 4\cdot-5 = 9x^2 - 3x - 20 $$

(iv) $\left(y^2+\frac{3}{2}\right)\left(y^2-\frac{3}{2}\right)$:

Using the second identity: $$ \left(y^2\right)^2 - \left(\frac{3}{2}\right)^2 = y^4 - \frac{9}{4} $$

(v) $(3-2x)(3+2x)$:

Again, using the second identity: $$ 3^2 - (2x)^2 = 9 - 4x^2 $$

So the products are:

(i) $x^2 + 14x + 40$

(ii) $x^2 - 80$

(iii) $9x^2 - 3x - 20$

(iv) $y^4 - \frac{9}{4}$

(v) $9 - 4x^2$

Evaluate the following products without multiplying directly:

(i) $103 \times 107$

(ii) $95 \times 96$

(iii) $104 \times 96$

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Factorise the following using appropriate identities:

(i) $9 x^{2}+6 x y+y^{2}$

(ii) $4 y^{2}-4 y+1$

(iii) $x^{2}-\frac{y^{2}}{100}$

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Expand each of the following, using suitable identities:

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

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

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

(iv) $(3 a-7 b-c)^{2}$

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

(vi) $\left[\frac{1}{4} a-\frac{1}{2} b+1\right]^{2}$

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Factorise:

(i) $4 x^{2}+9 y^{2}+16 z^{2}+12 x y-24 y z-16 x z$

(ii) $2 x^{2}+y^{2}+8 z^{2}-2 \sqrt{2} x y+4 \sqrt{2} y z-8 x z$

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Write the following cubes in expanded form:

(i) $(2 x+1)^{3}$

(ii) $(2 a-3 b)^{3}$

(iii) $\left[\frac{3}{2} x+1\right]^{3}$

(iv) $\left[x-\frac{2}{3} y\right]^{3}$

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Evaluate the following using suitable identities:

(i) $(99)^{3}$

(ii) $(102)^{3}$

(iii) $(998)^{3}$

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Factorise each of the following:

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

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

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

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

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

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

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

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Factorise each of the following:

(i) $27 y^{3}+125 z^{3}$

(ii) $64 m^{3}-343 n^{3}$

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

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

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If $x+y+z=0$, show that $x^{3}+y^{3}+z^{3}=3 x y z$.

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

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

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

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Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:

$\text { Area: } 25 a^{2}-35 a+12$

$\text { Area: } 35 y^{2}+13 y-12$

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

Volume : $3 x^{2}-12 x$

Volume : $12 k y^{2}+8 k y-20 k$