Question

Note: If p(a) = 0 then (x - a) is a factor of p(x)

given that f(-1), f(-2) are zeros of f(x) find the missing values of these coefficients p, q

f(x) = x³ + px² - qx - 4

After finding the values of the coefficients factor and find the solutions of x1, x2 and x3.​

Answer 1

Since f(-1) = 0 and f(-2) = 0, we can set up two equations:

(-1)³ + p(-1)² - q(-1) - 4 = 0

(-2)³ + p(-2)² - q(-2) - 4 = 0

Simplifying each equation, we get:

-1 + p - q - 4 = 0

-8 + 4p - 2q - 4 = 0

Simplifying further, we get:

p - q = 5

2p - q = 6

Solving for p and q, we can add the two equations together to eliminate q:

p - q + 2p - q = 5 + 6

3p - 2q = 11

Then, we can substitute the value of q from the first equation into this equation to solve for p:

3p - 2(q + 5) = 11

3p - 2q - 10 = 11

3p - 2q = 21

3p - 2(5 + p) = 21

p - 10 = 7

p = 17

Substituting this value of p into either equation for q, we get:

q = p - 5 = 12

Therefore, the coefficients are p = 17 and q = 12. To factor the expression, we can use synthetic division or long division. Using synthetic division, we get:

-1 | 1 17 -12 -4

|__ -1 -16 28

1 1 12

This gives us the factorization:

f(x) = (x + 1)(x² + x + 12)

To find the solutions, we can use the quadratic formula on the quadratic factor:

x = (-1 ± sqrt(1 - 4(1)(12))) / 2(1)

x = (-1 ± sqrt(1 - 48)) / 2

x = (-1 ± sqrt(-47)) / 2

x1 = (-1 + i(sqrt(47))) / 2

x2 = (-1 - i(sqrt(47))) / 2

Therefore, the solutions are x1 = (-1 + i(sqrt(47))) / 2, x2 = (-1 - i(sqrt(47))) / 2, and x3 = -1.

Explanation:

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