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geniusboy [140]
2 years ago
15

6. Consider the polynomial p(x) = x^2 + kr^2 +x+6. Find a value of k so that x+1 is a factor of P. Find all the zeros of P.​

Mathematics
1 answer:
Nikitich [7]2 years ago
6 0

If <em>x</em> + 1 is a factor of <em>p(x)</em> = <em>x</em>³ + <em>k</em> <em>x</em>² + <em>x</em> + 6, then by the remainder theorem, we have

<em>p</em> (-1) = (-1)³ + <em>k</em> (-1)² + (-1) + 6 = 0   →   <em>k</em> = -4

So we have

<em>p(x)</em> = <em>x</em>³ - 4<em>x</em>² + <em>x</em> + 6

Dividing <em>p(x)</em> by <em>x</em> + 1 (using whatever method you prefer) gives

<em>p(x)</em> / (<em>x</em> + 1) = <em>x</em>² - 5<em>x</em> + 6

Synthetic division, for instance, might go like this:

-1   |   1    -4     1     6

...   |         -1    5    -6

----------------------------

...   |   1    -5    6     0

Next, we have

<em>x</em>² - 5<em>x</em> + 6 = (<em>x</em> - 3) (<em>x</em> - 2)

so that, in addition to <em>x</em> = -1, the other two zeros of <em>p(x)</em> are <em>x</em> = 3 and <em>x</em> = 2

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