Use the factor theorem and tell whether the first expression is a factor of P ( x ). x + 1; P ( x ) = x ³ + 3 x ² - 3 x - 5 . (Hint: Write x + 1 as x - (-1).) A) yes B) no

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To use the factor theorem, we need to determine if P(-1) = 0. Let's substitute -1 for x in P(x): P(-1) = (-1)^3³ + 3(-1)^2² - 3(-1) - 5 = -³ + 3² + 2 We can see that P(-1) is not equal to 0. However, we can rewrite x + 1 as x - (-1), and then substitute -1 for x - (-1) in P(x): P(x - (-1)) = (x - (-1))^3³ + 3(x - (-1))^2² - 3(x - (-1)) - 5 = (x + 1)^3³ + 3(x + 1)^2² - 3(x + 1) - 5 = (x^3 + 3x^2 + 3x + 1)³ + (3x^2 + 6x + 3)² - 3x - 3 - 5 = x^3³ + 3x^2³ + 3x³ + ³ + 3x^2² + 6x² + 3² - 3x - 8 = x^3(³) + x^2(3³ + 3²) + x(3³ + 6² - 3) + (³ + 3² - 8) We can see that (x+1) is indeed a factor of P(x) because P(x-(-1)) is equal to (x+1) times another polynomial. Therefore, the answer is A.

Use the Factor Theorem and Tell Whether the First Expression