If x + 2 is a factor of the trinomial 4x² + kx-6, what is the value of k?
The value of k is -5.
To determine the value of k such that x + 2 is a factor of the trinomial 4x² + kx - 6, we can use the Factor Theorem. This theorem states that if x + 2 is a factor, then substituting -2 for x in the polynomial should yield zero.
Substituting -2 into the polynomial gives: 4(-2)² + k(-2) - 6 = 0. This simplifies to 16 - 2k - 6 = 0, leading to 10 - 2k = 0. Solving for k, we find k = -5.
If k = -3, substituting -2 into the polynomial yields: 4(-2)² + (-3)(-2) - 6 = 16 + 6 - 6 = 16, which does not equal zero. Thus, k = -3 does not satisfy the condition that x + 2 is a factor.
For k = 3, substituting -2 into the polynomial results in: 4(-2)² + 3(-2) - 6 = 16 - 6 - 6 = 4, which again does not equal zero. Therefore, k = 3 is not the correct value.
Substituting k = 5, we calculate: 4(-2)² + 5(-2) - 6 = 16 - 10 - 6 = 0. While this yields zero, it does not match the value of k we are looking for since the polynomial will not factor correctly with this k.
If k = 8, substituting -2 gives: 4(-2)² + 8(-2) - 6 = 16 - 16 - 6 = -6, which does not equal zero. Therefore, k = 8 is also incorrect.
Using the Factor Theorem, we found that substituting -2 into the trinomial allowed us to solve for k effectively. The only value of k that ensures x + 2 is a factor of the polynomial 4x² + kx - 6 is -5. The other options failed to satisfy the factor condition, reaffirming that k = -5 is the unique solution.
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