Question

If\ the\ polynomial\ p\left(x\right)\ =\ 2x^3\ +\ ax^{2\ }-\ 7x\ +b,\ p\left(1\right)\ =\ 3\ and\ p\left(2\right)\ =\ 19,\ find\ a\ \&\ b

pls tell fast

Answer 1

Value of a =3

Value of b =5

Explanation:

`If\ the\ polynomial\ p\left(x\right)\ =\ 2x^3\ +\ ax^(2\ )-\ 7x\ +b,\ p\left(1\right)\ =\ 3\ and\ p\left(2\right)\ =\ 19,\ find\ a\ \&\ b`

We are given p(1)=3 and p(2)=19

Putting x =1 to find p(1)

`p\left(x\right)\ =\ 2x^3\ +\ ax^(2\ )-\ 7x\ +b\\p(1)=2(1)^3+a(1)^2-7(1)+b\\We \ know \ p(1)=3\\3=2+a-7+b\\3=-5+a+b\\a+b=3+5\\a+b=8`

Now putting x=2 as p(2)=19

`p\left(x\right)\ =\ 2x^3\ +\ ax^(2\ )-\ 7x\ +b\\p(2)=2(2)^3+a(2)^2-7(2)+b\\We \ know \ p(2)=19\\19=2(8)+4a-14+b\\19=16+4a-14+b\\4a+b=19-16+14\\4a+b=17`

Solving these equations to find values of a and b

`a+b=8---eq(1)\\4a+b=17---eq(2)`

Subtract both equations

`a+b=8\\4a+b=17\\-\ \ \ - \ \ \ -\\------\\-3a=-9\\a=(-9)/(-3)\\a=3`

So, value of a =3

Now finding value of b bu putting value of a in equation 1

`a+b=8\\3+b=8\\b=8-3\\b=5`

So, Value of b =5