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 ≠0 $
(vii) $ p(x)=c x+d, c ≠0, c, d $ are real numbers.
To do:
We have to find the zeroes of the given polynomials.
Solution :
The zero of a polynomial is defined as any real value of $x$, for which the value of the polynomial becomes zero.
Therefore,
(i) Zero of the polynomial $p(x) = x+5$ is,
$x+5 = 0$
$x = -5$.
Zero of the polynomial $p(x) = x+5$ is $-5$.
(ii) Zero of the polynomial $p(x) = x-5$ is,
$x-5 = 0$
$x = 5$.
Zero of the polynomial $p(x) = x-5$ is $5$.
(iii) Zero of the polynomial $p(x) = 2x+5$ is,
$2x+5 = 0$
$2x = -5$
$x=\frac{-5}{2}$
Zero of the polynomial $p(x) = 2x+5$ is $\frac{-5}{2}$.
(iv) Zero of the polynomial $p(x) = 3x-2$ is,
$3x-2 = 0$
$3x = 2$
$x=\frac{2}{3}$
Zero of the polynomial $p(x) = 3x-2$ is $\frac{2}{3}$.
(v) Zero of the polynomial $p(x) = 3x$ is,
$3x = 0$
$x = 0$.
Zero of the polynomial $p(x) = 3x$ is $0$.
(vi) Zero of the polynomial $p(x) = ax$ is,
$ax = 0$
$x = \frac{0}{a}$
$x=0$
Zero of the polynomial $p(x) = ax$ is $0$.
(vii) Zero of the polynomial $p(x) = cx+d$ is,
$cx+d = 0$
$cx = -d$
$x=\frac{-d}{c}$
Zero of the polynomial $p(x) = cx+d$ is $\frac{-d}{c}$.
Related Articles
- 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) \( p(x)=2 x+1, x=\frac{1}{2} \)
- 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) \( p(x)=k x^{2}-3 x+k \)
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:(i) $p(x) = x^3 - 3x^2 + 5x -3, g(x) = x^2-2$(ii) $p(x) =x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 -x$(iii) $p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- 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) \( p(x)=x^{3}-4 x^{2}+x+6, g(x)=x-3 \)
- 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) \( p(x)=(x-1)(x+1) \)
- divide the polynomial $p( x)$ by the polynomial $g( x)$ and find the quotient and remainder in each of the following: $( p(x)=x^{3}-3 x^{2}+5 x-3$, $g(x)=x^{2}-2$.
- The degree of polynomial in $ P(x)=5 x^{3}+4 x^{2}+7 x $A) 1B) 2C) 3D) 0
- Find the zeros of the polynomial \( x^{2}+x-p(p+1) \).
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:$p(x) =x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 -x$
- For which values of \( a \) and \( b \), are the zeroes of \( q(x)=x^{3}+2 x^{2}+a \) also the zeroes of the polynomial \( p(x)=x^{5}-x^{4}-4 x^{3}+3 x^{2}+3 x+b \) ? Which zeroes of \( p(x) \) are not the zeroes of \( q(x) \) ?
- Find the values of $x$ in each of the following:\( 2^{5 x} \p 2^{x}=\sqrt[5]{2^{20}} \)
- Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder, in each of the following:$p(x) = x^4 - 5x + 6, g(x) = 2 -x^2$
- Answer the following and justify:What will the quotient and remainder be on division of \( a x^{2}+b x+c \) by \( p x^{3}+q x^{2}+r x+s, p ≠0 ? \)
- Solve the following equation$x^{2}r^{2}+2r(2q-p) x+(p-2q)^{2}=0$
- Find the zero of the polynomial:$P(x)=x- 5x+6$
Kickstart Your Career
Get certified by completing the course
Get Started