Question

if the zeros of a cubic polynomial x cube minus 6 X square + 3 X + 10 hour of the form a a + b and a + 2B for some real numbers A and B find the values of A and B as well as zeros of the given polynomial ​

Answer 1

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Given cubic polynomial is

p(x)=x³−6x²+3x+10

The zeros of the polynomial p(x) are of the form a, a+b and a+2b

Then,

a+a+b+a+2b=− −6/1

=>3a+3b=6

=>a+b=2 ----------------(i)

Also, a(a+b)+(a+b)(a+2b)+a(a+2b)= 3/1

=>a²+ab+a ² +2ab+ab+2b²+a²+2ab=3

=>3a ² +2b ² +6ab=3 ----(ii)and

a(a+b)(a+2b)=−10/1

=>a³+a ² b+2a²b+2ab²=10

From (i), b=2−a

Putting this value in (ii), we get

=>3a ²+2(2−a) ²+6a(2−a)=3

=>3a ² +2(4−4a+a ² )+12a−a ²=3

=>−a ²+4a+5=0

=>a ² −4a−5=0

=>a² −5a+a−5=0

=>a(a−5)+(a−5)=0

=>.(a−5)(a+1)=0

=>a=5 or a=−1

=>b=−3 or b=3 respectively

From equation (iii), we get

at a=5, b=−3

=>5 ³+(5) ²(−3)+2(5) ²(−3)+2(5)(−3) ²

=−10

=>125−75−150+90=−10

=>−10=−10 which is true.

and at a=−1, b=3, we have

=>(−1) ³ +(−1) ² 3+2(−1) ² (3)+2(−1)(3)²

=−10

=>−1+3−6−18=10

=>−22=−10 which is not true.

Thus, a=5, b=−3

Zeros of the polynomial are 5, 5−3, 5−2×3 ie 5,2,−1

Answer 2

Answer:

here it is hope it's helpful for you .

Step-by-step explanation:

x³-6x²+3x+10

sum of the zeroes = -b/a

a + a+b + a+2b = -(-6) / 1

3a +3b = 6

3(a+b) = 6

a+b = 2. 1eq.

product of zeroes = c/a

a(a+b) + (a+b)(a+2b) + (a+2b)a = 3/1

a²+ab+ a²+ 2b² + 2ab +ab + a² +2ab = 3

3a² + 2b² + 6ab = 3

from 1 eq we can say that a = 2-b

3(2-b)² + 2b² +6(2-b)b = 3

3(2²+b²-4b) +2b² + 6(2b-b²) = 3

3(4+b²-4b) +2b² + 6(2b-b²) = 3

12 + 3b² -12b + 2b² +12b -6b² = 3

-b²+12 = 3

-b² = -9

b = 3 hence we can find a from 1eq.

a+b = 2

a+3 = 2

a= -1

so the zeroes are

a= -1

a+b = -1+3 = 2

a+2b = -1 +2(3) = 5