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The polynomial function f is defined as f(m)=(m^(3)-m^(2)-17 m-15)(m+1). When f(m) is divided by (m+1), what is the remainder?

The polynomial function f f is defined as f(m)=(m3−m2−17m−15)(m+1) f(m)=\left(m^{3}-m^{2}-17 m-15\right)(m+1) . When f(m) f(m) is divided by (m+1) (m+1) , what is the remainder?

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Full solution

Q. The polynomial function f f is defined as f(m)=(m3−m2−17m−15)(m+1) f(m)=\left(m^{3}-m^{2}-17 m-15\right)(m+1) . When f(m) f(m) is divided by (m+1) (m+1) , what is the remainder?
  1. Apply Remainder Theorem: To find the remainder when f(m)f(m) is divided by (m+1)(m+1), we can use the Remainder Theorem which states that the remainder of the division of a polynomial f(x)f(x) by (x−c)(x - c) is f(c)f(c).
  2. Substitute mm with −1-1: In this case, we need to find f(−1)f(-1) because we are dividing by (m+1)(m+1), which is the same as (m−(−1))(m - (-1)).
  3. Simplify the expression: Now, let's substitute mm with −1-1 in the polynomial f(m)=(m3−m2−17m−15)(m+1)f(m) = (m^3 - m^2 - 17m - 15)(m + 1).
  4. Simplify the expression: Now, let's substitute mm with −1-1 in the polynomial f(m)=(m3−m2−17m−15)(m+1)f(m) = (m^3 - m^2 - 17m - 15)(m + 1).f(−1)=((−1)3−(−1)2−17(−1)−15)(−1+1)f(-1) = ((-1)^3 - (-1)^2 - 17(-1) - 15)(-1 + 1).
  5. Simplify the expression: Now, let's substitute mm with −1-1 in the polynomial f(m)=(m3−m2−17m−15)(m+1)f(m) = (m^3 - m^2 - 17m - 15)(m + 1).f(−1)=((−1)3−(−1)2−17(−1)−15)(−1+1)f(-1) = ((-1)^3 - (-1)^2 - 17(-1) - 15)(-1 + 1).Simplify the expression: f(−1)=((−1)−(1)+17−15)(0)f(-1) = ((-1) - (1) + 17 - 15)(0).
  6. Simplify the expression: Now, let's substitute mm with −1-1 in the polynomial f(m)=(m3−m2−17m−15)(m+1)f(m) = (m^3 - m^2 - 17m - 15)(m + 1).f(−1)=((−1)3−(−1)2−17(−1)−15)(−1+1)f(-1) = ((-1)^3 - (-1)^2 - 17(-1) - 15)(-1 + 1).Simplify the expression: f(−1)=((−1)−(1)+17−15)(0)f(-1) = ((-1) - (1) + 17 - 15)(0).f(−1)=(0)(0)=0f(-1) = (0)(0) = 0.

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