The expression (y^(2)-t^(2))(y+k) can be written as y^(3)+36y^(2)-9y+s where t,k, and s are constants. What is the value of s ? Choose 1 answer: (A) -324 (B) -54 (c) 54 (D) 324

Question

The expression  (y^(2)-t^(2))(y+k) can be written as  y^(3)+36y^(2)-9y+s where  t,k, and  s are constants. What is the value of  s ? Choose 1 answer: (A) -324 (B) -54 (c) 54 (D) 324

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Expand and Compare: We need to expand the expression (y^2 - t^2)(y + k) and compare it to the given expression y^3 + 36y^2 - 9y + s to find the value of s.Factor Difference of Squares: First, recognize that y^2 - t^2 is a difference of squares and can be factored into (y + t)(y - t).Distribute Across Binomial: Now, distribute (y + t)(y - t) across (y + k) to get the expanded form. Start with the first term of the first binomial (y + t): (y + t)(y^2 - t^2) = y(y^2 - t^2) + t(y^2 - t^2).Expand y(y^2 - t^2): Expanding y(y^2 - t^2) gives us y^3 - yt^2.Expand t(y^2 - t^2): Expanding t(y^2 - t^2) gives us ty^2 - t^3.Add k(y^2 - t^2): Now, add k(y^2 - t^2) to the expression: y^3 - yt^2 + ty^2 - t^3 + k(y^2 - t^2).Combine Like Terms: Expanding k(y^2 - t^2) gives us ky^2 - kt^2.Compare Coefficients: Combine like terms to get the full expanded expression: y^3 + (ty^2 + ky^2) - yt^2 - t^3 - kt^2.Solve for t: Now, we compare the coefficients of the expanded expression to the given expression y^3 + 36y^2 - 9y + s. We see that the coefficient of y^2 in the expanded expression is t + k, which must be equal to 36 in the given expression.Substitute t into Equation: We also see that the coefficient of y in the expanded expression is -yt, which must be equal to -9 in the given expression. This gives us the equation -yt = -9.Find Value of s: From the equation -yt = -9, we can solve for t by dividing both sides by -y (assuming y is not zero). This gives us t = 9/y.Compare Constant Terms: Now, we substitute t = 9/y into the equation t + k = 36 to find the value of k. This gives us 9/y + k = 36.Since we cannot solve for k without knowing the value of y, we look at the constant terms in the expanded expression and the given expression. The constant term in the expanded expression is -t^3 - kt^2, which must be equal to s in the given expression.We know that t = 9/y, so we substitute this into the constant term -t^3 - kt^2 to find s. This gives us -t^3 - kt^2 = -(9/y)^3 - k(9/y)^2.However, we made a mistake in the previous step. We do not need to substitute t = 9/y into the constant term because we are looking for the value of s, which is the constant term in the given expression y^3 + 36y^2 - 9y + s. We should directly compare the constant terms from the expanded expression and the given expression.

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