Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method.

Question

g(x)=2x^6+2x^4-x²+1

(a) g(2)=?

(b) g(1)=?

(c) g(3)=?

(d) g(-1)=?

Answer 1

for synthetic division method

first rewrite the polynomial indicating all powers of the variable

g(x) = 2x^6 + 2x^4 - x^2 + 1 = 2X^6 + 0x^5 + 2x^4 + 0x^3 - X^2 +0x +1

now do synthetic division with all t coeefficients in the rewritten form of the polynomial

(a) g(2) = the remainder of g(x) divided by (x-2)

2 | 2 0 2 0 -1 0 1

_ 4_8_20 40_78_156____

2 4 10 20 39 78 157

since the remainder is 157 the value of g(2) = 157

(b) g(1) = the remainder of g(x) divided by (x-1)

1 | 2 0 2 0 -1 0 1

__2_2_4__4_3 3__

2 2 4 4 3 3 4

since the remainder is 4 the value of g(1) = 4

the procedure is the same for the remaining problems