Zeros of a polynomial

Q.

Find a quadratic polynomial whose zeroes are $2$ and $-5$ respectively.

Q. If $\alpha$ and $\beta$ are the zeros of polynomial
$x^2+3x-2$, find $\frac{1}{(\alpha {)}^3}+\frac{1}{(\beta {)}^3}$.

1. $\frac{8}{45}$
2. $\frac{-8}{45}$
3. $\frac{45}{8}$
4. $\frac{-45}{8}$

Q.

If $\alpha$ and $\beta$ are the zeros of a polynomial
$f\left(x\right)=6x^2+x-2$, find the values of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$.

1. $-\frac{25}{12}$

2. $\frac{25}{12}$

3. $\frac{25}{36}$

4. $\frac{12}{25}$

Q. Find the zero of the polynomial in each of the following cases:
(i) $p\left(x\right)=x+5$
(ii) $p\left(x\right)=x-5$
(iii) $p\left(x\right)=2x+5$
(iv) $p\left(x\right)=3x-2$
(v) $p\left(x\right)=3x$
(vi) $p\left(x\right)=ax,a\ne 0$
(vii) $p\left(x\right)=cx+d,c\ne 0,c,d$ are real numbers.

Q. To find zeros of any polynomial function in x i.e. f(x), we use f(x)___0. Which operator should come in the blank?

1. =
2. >
3. <
4. $\ne$

Q. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) $p\left(x\right)=3x+1,x=-\frac{1}{3}$
(ii) $p\left(x\right)=5x-\pi ,x=\frac{4}{5}$
(iii) $p\left(x\right)=x^2-1,x=1,-1$
(iv) $p\left(x\right)=(x+1)(x-2),x=-1,2$
(v) $p\left(x\right)=x^2,x=0$
(vi) $p\left(x\right)=lx+m,x=-\frac{m}{l}$
(vii) $p\left(x\right)=3x^2-1,x=-\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}}$
(viii) $p\left(x\right)=2x+1,x=\frac{1}{2}$

Q. Find the zeroes of f(x) = 18x - 72.

1. 3
2. 6
3. 4
4. 2

Q. Question 11
Find the zeroes of the polynomial in each of the following.

(i) p(x) = x – 4

(ii) g(x) = 3 – 6x

(iii) q(x) = 2x – 7

(iv) h(y) = 2y

Q.

$\text{Find a quadratic polynomial, the sum and product of whose zeroes are}-3\text{and}2\text{respectively.}$

Q.

Question 2 (iv)
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

1, 1

Q.

Verify whether the following values indicated against them are zeroes of the polynomial.
$\left(i\right)p\left(x\right)=5x-\pi ,x=\frac{4}{5}$
$\left(ii\right)p\left(x\right)=x^2-1,x=1,-1$ [4 MARKS]

Q. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x + 1, x = $-\frac{1}{3}$
(ii) $p\left(x\right)=5x-\pi ,x=\frac{4}{5}$
(iii) $p\left(x\right)=x^2-1,x=1,-1$
(iv) p(x) = (x +1) (x -2), x = - 1, 2

Q. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x + 1, x = $-\frac{1}{3}$
(ii) $p\left(x\right)=5x-\pi ,x=\frac{4}{5}$
(iii) $p\left(x\right)=x^2-1,x=1,-1$
(iv) p(x) = (x +1) (x -2), x = - 1, 2

Q. If one zero of the polynomial $3x^2-8x+2k+1$ is seven times the other, then the value of $k$ is ___.

1. $-\frac{1}{2}$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $-\frac{2}{3}$

Q.

Find all the common zeroes of the polynomials $x^3+5x^2-9x-45$ and $x^3+8x^2+15x$.

1. 3

2. 5

3. -3

4. -5

Q.

If $\alpha ,\beta$ and $\gamma$ are the zeroes of the polynomial $f\left(x\right)=a{x}^3+b{x}^2+cx+d$, then $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }$ is _____.

1. $\frac{c}{d}$

2. $\frac{a}{d}$

3. $-\frac{c}{d}$

4. $-\frac{b}{a}$

Q.

If $\alpha ,\beta$ are the zeroes of the polynomial $x^2-px+36$ and ${\alpha }^2$+${\beta }^2$ = 9, then what is the value of p?

1. ±9

2. ±6

3. ±3

4. ±8

Q.

If $p\left(x\right)=x^2-3\sqrt{3}x+1$, then the value of $p\left(3\sqrt{3}\right)$ is 0.

1. False

2. True

Q. $\text{A quadratic polynomial whose zeroes}\text{are}\frac{3}{4}\text{and}\frac{1}{2}\text{is}$

1. $\frac{8x^2-10x+3}{8}$
2. $\frac{10x^2-8x+3}{4}$
3. $\frac{8x^2+10x+3}{8}$
4. $\frac{10x^2+8x+3}{4}$

Q.

$\text{Find the zeroes of the polynomial}{x}^2-2.$

1. $\sqrt{2}\text{and}-\sqrt{5}$

2. $4\text{and}3$

3. $\sqrt{2}\text{and}-\sqrt{2}$

4. $-5\text{and}2$

Q.

$\text{Zeroes of}{x}^2-56x+108\text{are}$______.

1. both positive

2. both negative

3. both equal

4. one negative and one positive

Q.

The zero of p(x) = 5x - 75 is

___

Q. The zeroes of $mn(x^2+1)=(m^2+n^2)x$ are:

1. $\frac{m}{n}and\frac{n}{m}$
2. $\frac{-m}{n}and\frac{n}{m}$
3. $\frac{-m}{n}and\frac{-n}{m}$
4. $\frac{m}{n}and\frac{-n}{m}$

Q. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x + 1, x = $-\frac{1}{3}$
(ii) $p\left(x\right)=5x-\pi ,x=\frac{4}{5}$
(iii) $p\left(x\right)=x^2-1,x=1,-1$
(iv) p(x) = (x +1) (x -2), x = - 1, 2