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
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Answer 1

Answer:

Value of a =3

Value of b =5

Step-by-step 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

$a+b=8\\4a+b=17\\-\ \ \ - \ \ \ -\\------\\-3a=-9\\a=\frac{-9}{-3}\\a=3$

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