Question

If the product of two zeroes of the polynomial fx=2x3+6x2-4x+9 is 3, then its third zero is


a. 92
b. -32
c. 32
d. -92


If the sum of the squares of the zeroes of the quadratic polynomial fx=x2-8x+kis 40, find the value of k.
a.12
b. -12
c. 14
d. -14

Answer 1

Answer:

1. b-32 2. 12 pleassss mark me as brainleast

Answer 2

Answer:

1) Option b (not -32 it is -3/2)

2) Option a

Step-by-step explanation:

Solutions:-

1)

Given Polynomial f(x) = 2x³+6x²-4x+9

On Comparing this with the standard Cubic Polynomial ax³+bx²+cx+d

a = 2

b = 6

c = -4

d = 9

Since it is a Cubic Polynomial then it has at most three zeroes

Let the three zeroes be A,B and C

We know that

Product of the zeroes = -d/a

=> A×B×C = -9/2 -----(1)

Given that

Product of two zeroes = 3

Let A×B = 3

Now, (1) becomes

=> 3×C = -9/2

=>C = (-9/2)/3

=> C = -9/(2×3)

=> C = -9/6

=> C = -3/2

Answer:-

The third Zero of the given Polynomial is -3/2

2)

Given Polynomial f(x) = x²-8x+k

On Comparing this with the standard Quadratic Polynomial ax²+bx+c

a = 1

b = -8

c = k

Let the three zeroes be A,B

We know that

Sum of the zeroes = -b/a

=> A+B = -(-8)/1

=> A+B = 8 ---------(1)

Product of the zeroes = c/a

=> A×B= k/1

=> AB = k --------(2)

On squaring both sides of equation (1) then

=> (A+B)² = 8²

=> A²+B²+2AB = 64-----(3)

Given that

The sum of the squares of the zeroes = 40

=> A²+B² = 40 --------(4)

Now,(4) becomes

=> 40+2k = 64

=> 2k = 64-40

=> 2k = 24

=> k = 24/2

=> k = 12

Answer:-

The value of k for the given problem is 12

Used formulae:-

• The standard quadratic polynomial is ax²+bx+c
• Sum of the zeores = -b/a
• Product of the zeroes = c/a
• The standard Cubic Polynomial ax³+bx²+cx+d
• Sum of the zeroes = -b/a
• Product of the zeroes = -d/a
• Sum of the product of the two zeroes taken at a time = c/a