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Marysya12 [62]
2 years ago
8

Find the value of k such that x-2 is a factor of 3x³-kx²+5x+k​

Mathematics
1 answer:
Nuetrik [128]2 years ago
6 0

Answer:

\displaystyle k = \frac{34}{3}

Step-by-step explanation:

We are given the polynomial:

\displaystyle P(x) = 3x^3 - kx^2 + 5x + k

And we want to determine the value of <em>k</em> such that (<em>x</em> - 2) is a factor of the polynomial.

Recall that the Factor Theorem states that a binomial (<em>x</em> - <em>a</em>) is a factor of a polynomial P(x) if and only if P(a) = 0.

Our binomial factor is (<em>x</em> - 2). Thus, <em>a</em> = 2.

Hence, by the Factor Theorem, P(2) must equal zero.

Find P(2):

\displaystyle \begin{aligned} P(2) &= 3(2)^3 - k(2)^2 + 5(2) + k \\ \\ &= 3(8) - 4k + 10 + k \\ \\ &= 34 - 3k  \end{aligned}

This must equal zero. Hence:

\displaystyle \begin{aligned} 34 - 3k &= 0 \\ \\ -3k &= -34 \\ \\ k = \frac{34}{3}  \end{aligned}

In conclusion, <em>k</em> = 34/3.

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