Answer: Read bottom
Step-by-step explanation:
To find P(c) where P(x) = 2x^2 + 5x + 5 and c = -3, we can use direct substitution and the remainder theorem.
(a) Direct Substitution:
To find P(c) using direct substitution, we substitute -3 for x in the polynomial P(x).
P(c) = 2(-3)^2 + 5(-3) + 5
= 2(9) - 15 + 5
= 18 - 15 + 5
= 8
Therefore, P(-3) = 8.
(b) Remainder Theorem:
The remainder theorem states that if we divide a polynomial P(x) by x - c, then the remainder will be P(c).
To find P(c) using the remainder theorem, we can divide P(x) by x - c and evaluate the remainder at c.
In this case, c = -3, so we divide P(x) = 2x^2 + 5x + 5 by x - (-3), which is x + 3.
Using long division or synthetic division, we can find the remainder:
2x + 1
---------------
x + 3 | 2x^2 + 5x + 5
- (2x^2 + 6x)
--------------
-x + 5
The remainder is -x + 5.
Evaluating the remainder at c = -3, we get P(-3) = -(-3) + 5 = 8.
Therefore, P(-3) = 8, which matches the result obtained using direct substitution.
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