✰Answer:
p(-3) = 152.
✰Step-by-step explanation:
To find p(c) for the polynomial p(x) = 2x^4 + 2x^2 + 5x + 5, where c = -3, we can use direct substitution and the remainder theorem.
(a) Direct Substitution:
To find p(-3) by direct substitution, we substitute -3 in place of x in the polynomial and calculate the result.
Substituting -3 for x in p(x), we get:
p(-3) = 2(-3)^4 + 2(-3)^2 + 5(-3) + 5
Simplifying the expression, we have:
p(-3) = 2(81) + 2(9) - 15 + 5
p(-3) = 162 + 18 - 15 + 5
p(-3) = 170
Therefore, p(-3) = 170.
(b) Remainder Theorem:
The remainder theorem states that if we divide a polynomial by x - c, the remainder will be equal to p(c). In this case, c = -3.
Using the remainder theorem, we can divide the polynomial p(x) by x - c, which is x - (-3), or x + 3.
Performing the division, we get:
2x^4 + 2x^2 + 5x + 5 ÷ (x + 3)
-6x^3 + 18x^2 - 49x + 152
_______________________________
x + 3 | 2x^4 + 0x^3 + 2x^2 + 5x + 5
Since the remainder is the last term in the division, p(-3) = 152.
Therefore, p(-3) = 152.
✰
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