x^(3)+7x^(2)-36 The polynomial has zeros at -6 and 2 . If the remaining zero is z, then what is the value of -z ?

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x^(3)+7x^(2)-36 The polynomial has zeros at -6 and 2 . If the remaining zero is  z, then what is the value of  -z ?

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Factor Theorem: We know that the polynomial x^3 + 7x^2 - 36 has zeros at -6 and 2. Since polynomials are continuous and differentiable everywhere, we can use the Factor Theorem to express the polynomial as a product of its factors. The Factor Theorem states that if r is a root of the polynomial, then (x - r) is a factor of the polynomial.Finding Factors: First, we can write down the factors corresponding to the known zeros -6 and 2. This gives us (x + 6) and (x - 2) as factors of the polynomial.Expanding Factors: Now, we can express the polynomial as the product of its factors and the unknown factor (x - z), where z is the remaining zero. The polynomial can be written as (x + 6)(x - 2)(x - z).Coefficient Matching: To find the value of z, we need to expand the known factors and set them equal to the given polynomial x^3 + 7x^2 - 36. Let's expand (x + 6)(x - 2) first. (x + 6)(x - 2) = x^2 - 2x + 6x - 12 = x^2 + 4x - 12Equating Constant Terms: Next, we multiply the expanded expression (x^2 + 4x - 12) by (x - z) and set it equal to the given polynomial x^3 + 7x^2 - 36. However, we can also use the fact that the coefficients of the x^2 term in the polynomial must match. The coefficient of x^2 in the given polynomial is 7, and we already have a coefficient of 1 from (x - z), so the coefficient from (x^2 + 4x - 12) must be 6.Solving for z: Since we know that the coefficient of x^2 must be 7, and we have x^2 from (x - z), the remaining coefficient from (x^2 + 4x - 12) must be 6. This means that the coefficient of x^2 in (x^2 + 4x - 12) is already correct, and we do not need to adjust the factor (x - z). Therefore, we can conclude that the polynomial (x + 6)(x - 2)(x - z) is equivalent to the given polynomial x^3 + 7x^2 - 36.Now, we can equate the constant terms from the expanded form of (x + 6)(x - 2)(x - z) to the constant term of the given polynomial, which is -36. The constant term from (x + 6)(x - 2) is -12, so when we multiply by (x - z), the constant term must be -36. -12 * (-z) = -36Solving for z, we divide both sides by -12. -z = -36 / -12 -z = 3

$x^{3}+7 x^{2}-36$ $\newline$ The polynomial has zeros at $-6$ and $2$ . If the remaining zero is $z$ , then what is the value of $-z$ ?

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$x^{3}+7 x^{2}-36$ $\newline$ The polynomial has zeros at $-6$ and $2$ . If the remaining zero is $z$ , then what is the value of $-z$ ?