Question

Find the quadratic polynomial whose zeroes are -2 and -5. Verify the relationship between zeroes and coefficient of the polynomial.

Answer 1

Answer:

The quadratic polynomial is x²+7 x+ 10

Step-by-step explanation:

Given,

-2 and -5 are zeros of the quadratic polynomial.

To find,

1. The quadratic polynomial
2. Verify the relationship between zeroes and the coefficient of the polynomial

Recall the concept

• If the roots of the quadratic equation are given then the quadratic equation is x²- (sum of roots) x+ (product of roots) = 0

• The relation between the zeroes and coefficients of a quadratic equation ax² + bx+c = 0 are

        Sum of zeroes =  $\frac{-b}{a}$ and  Product of roots = $\frac{c}{a}$

Solution:

Since -2 and -5 are the roots, then sum of roots = -2+-5 = -7

Product of roots = (-2)(-5) = 10

∴ The required equation is x²- (sum of roots) x+ (product of roots) = 0

⇒ x²- (-7) x+ 10 = 0

⇒ x²+7 x+ 10 = 0

∴ The quadratic polynomial is x²+7 x+ 10

Comparing this equation with ax² + bx+c = 0, we get

a = 1, b=7 and c=12

$\frac{-b}{a}$  = -7 = (-5)+ (-2) =  Sum of zeros

$\frac{c}{a}$ = 10 = (-5)(-2) = Product of zeroes

Hence the relation between the coefficients and zeroes, that is

Sum of zeroes =  $\frac{-b}{a}$ and  Product of roots = $\frac{c}{a}$  is verified

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Answer 2

Step 1: Given data

zeros of the quadratic polynomial$=-2,-5$

quadratic polynomial$=?$

We also have to verify the relationship between zeroes and the coefficient of the polynomial.

Step 2: Using the concept

1. If the roots of the quadratic equation are given then,               quadratic equation  $=x^{2} - (sum\ of\ roots)x+ (product\ of\ roots) = 0$
2. The relation between the zeroes and coefficients of a quadratic equation $ax^{2} + bx+c = 0$ are,

• Sum of zeroes $=\frac{-b}{a}$ 
• Product of roots$=\frac{c}{a}$

Step 3: Calculating the quadratic polynomial

Since $-2$ and $-5$ are the roots then,

sum of roots $= -2+-5 = -7$

Product of roots $= (-2)\times(-5) = 10$

∴ The required equation is,

$x^{2} - (-7) x+ 10 = 0$

$x^{2} +7 x+ 10 = 0$

Step 4: Verifying the relationship between zeroes and coefficient of the polynomial

Comparing the equation $x^{2} +7 x+ 10 = 0$ with $ax^{2} + bx+c = 0$

we get,

$a = 1, b=7,c=10$

sum of zeroes $=\frac{-b}{a}=\frac{-7}{1}=-7$  

product of zeroes $=\frac{c}{a} =\frac{10}{1} =10$

Hence, the quadratic polynomial is $x^{2} +7 x+ 10 = 0$ and the relation between the coefficients and zeroes is verified.

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