Question

find the zeroes of the following polynomial and verify the realtionship between the zeroes and the coefficients 1) x2-11x+30​

Answer 1

Answer:

Step-by-step explanation:

We first find the roots of the above quadratic equation: by splitting the middle term:

$x^2-11x+30=0\\\\x-6x-5x+30=0\\\\x(x-6)-5(x-6)=0\\\\(x-5)(x-6)=0\\\\x=5, 6$

Let α = -5 and β = 6

Now, we will verify the relationship between zeroes and coefficient of polynomial:

$\alpha +\beta =\dfrac{-b}{a}\\=\dfrac{-11}{1}\\\\=-11$

Similarly,

$\alpha\beta =\dfrac{c}{a}\\\\5\times 6=\dfrac{30}{1}\\\\30=30$

Hope this helps...

Answer 2

$\underline\mathfrak{Given:-}$

• x² - 11x + 30

$\underline\mathfrak{To \: \: Find:-}$

• Find the zeroes of the following polynomial and verify the realtionship between the zeroes and the coefficients ......?

$\underline\mathfrak{Solutions:-}$

• $\: \: \: \: \: P \: {(x)} \: \: = \: \: {x}^{2} \: - \: {11x} \: + \: {30}$

$\: \: \: \: \: \leadsto \: \: {x}^{2} \: - \: {11x} \: + \: {30} \: \: = \: \: {0}$

$\: \: \: \: \: \leadsto \: \: {x}^{2} \: - \: {6x} \: - \: {5x} \: + \: {30} \: \: = \: \: {0}$

$\: \: \: \: \: \leadsto \: \: {x} \: {({x} \: - \: {6})} \: - \: {({x} \: - \: {6})}$

$\: \: \: \: \: \leadsto \: \: {({x} \: - \: {5})} \: \: \: {({x} \: - \: {6})}$

$\: \: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: {5} \: \: \: and \: \: \: {x} \: \: = \: \: {6}$

$\: \: \: \: \: \: \: \: \: {\alpha} \: \: = \: \: {5} \: \: \: and \: \: \: {\beta} \: \: = \: \: {6}$

$\underline\mathfrak{Verification:-}$

x² - 11x + 30

• a = 1
• b = -11
• c = 30

$\: \: \: \: \: \therefore {Sum \: \: of \: \: zeroes} \: \: = \: \: \frac{ \: - \: coefficient \: \: of \: \: x}{coefficient \: \: of \: \: {x}^{2}}$

$\: \: \: \: \: \leadsto \: \: {\alpha} \: + \: {\beta} \: \: = \: \: \frac{-b}{a}$

$\: \: \: \: \: \leadsto \: \: {5} \: + \: {6} \: \: = \: \: - \: \frac{(-11)}{1}$

$\: \: \: \: \: \leadsto \: \: {11} \: \: = \: \: {11}$

$\: \: \: \: \: \therefore {Product \: \: of \: \: zeroes} \: \: = \: \: \frac{constant \: \: term}{coefficient \: \: of \: \: {x}^{2}}$

$\: \: \: \: \: \leadsto \: \: {\alpha} \: {\beta} \: \: = \: \: \frac{c}{a}$

$\: \: \: \: \: \leadsto \: \: {5} \: \times \: {6} \: \: = \: \: {\frac{30}{1}}$

$\: \: \: \: \: \leadsto \: \: {30} \: \: = \: \: {30}$

Verified.