Question

if p(x) =ax7+bx5+cx3+3 and p(7)=2,p(-7)=?​

Answer 1

p(x) = ax⁷ + bx⁵ + cx³ + 3

Now, put -x in place of x

p(-x) = a(-x)⁷ + b(-x)⁵ + c(-x)³ + 3

= -ax⁷ - bx⁵ - cx³ + 3

= -ax⁷ - bx⁵ - cx³ - 3 + 6

= -(ax⁷ + bx⁵ + cx³ + 3) + 6

Notice that the term in bracket is actually p(x)!!

→ p(-x) = -p(x) + 6

In the question p(7) is given

so finding p(-7) is a piece of cake now

p(-7) = -p(7) + 6

→ p(-7) = -2 + 6 = 4

Hence the answer is four.

Answer 2

Step-by-step explanation:

Consider the given function.

f(x)=ax^7+bx^3+cx−5

Since, f(−7)=7

So,

f(−7) = a(−7)^7 + b(−7)^3 + c(−7) − 5=7

−7^7xa − 7^3xb − 7c − 5 = 7

−(7^7xa + 7^3xb + 7c + 5) = 7

7^7xab + 7^3xbb+ 7c + 5 = −7

7^7xa + 7^3xb + 7c = -12. ..................(1)

Put x=7

f(7)=a(7)^7 + b(7)^3 + c(7) - 5

f(7)=7^7 x a + 7^3xb + 7c - 5

f(7)=−12−5 from equation (1)

f(7)=−17

Hence, this is the answer.