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nadya68 [22]
2 years ago
7

Find a polynomial of degree 3 with real coefficients and zeros of -3,-1, and 4, for which f(-2)=-24

Mathematics
1 answer:
Zanzabum2 years ago
4 0

We want to find a polynomial

<em>f(x)</em> = <em>a</em> <em>x</em>³ + <em>b</em> <em>x</em>² + <em>c</em> <em>x</em> + <em>d</em>

such that the roots of <em>f</em> are <em>x</em> = -3, <em>x</em> = -1, and <em>x</em> = 4, and <em>f(x)</em> takes on a value of -24 when <em>x</em> = -2.

The factor theorem for polynomials tells us that we can factorize <em>f(x)</em> as

<em>a</em> <em>x</em>³ + <em>b</em> <em>x</em>² + <em>c</em> <em>x</em> + <em>d</em> = <em>a</em> (<em>x</em> + 3) (<em>x</em> + 1) (<em>x</em> - 4)

Expand the right side:

(<em>x</em> + 3) (<em>x</em> + 1) (<em>x</em> - 4) = <em>x</em>³ - 13<em>x</em> - 12

So we have

<em>a</em> <em>x</em>³ + <em>b</em> <em>x</em>² + <em>c</em> <em>x</em> + <em>d</em> = <em>a x</em>³ - 13<em>a</em> <em>x</em> - 12<em>a</em>

<em />

In order for both sides to be equal, the coefficients of both polynomials on terms of the same degree must be equal. This means

<em>a</em> = <em>a</em> (of course)

<em>b</em> = 0 (there is no <em>x</em>² term on the right)

<em>c</em> = -13<em>a</em>

<em>d</em> = -12<em>a</em>

<em />

We also have that <em>f</em> (-2) = -24, which means

<em>f</em> (-2) = <em>a</em> (-2 + 3) (-2 + 1) (-2 - 4)

-24 = 6<em>a</em>

<em>a</em> = -4

which in turn tells us that <em>c</em> = 52 and <em>d</em> = 48.

So we found

<em>f(x)</em> = -4<em>x</em>³ + 52<em>x</em> + 48

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Which equation has a solution of -2?
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Solution:

I'm not sure if there is law for it or not, but you can replace each x to -2 if both sides are equal then the equation is true.

7x - 4 = 6x + 5\\7(-2) - 4 = 6(-2) + 5\\-14-4=-12+5\\-18\neq -7

A. is incorrect.

4x + 3 = 8x - 5\\4(-2) + 3 = 8(-2) - 5\\-8+3=-16-5\\-5\neq -21

B. is incorrect.

6x + 4 = 3x + 10\\6(-2) + 4 = 3(-2) + 10\\-12+4=-6+10\\-8\neq 4

C. is incorrect.

-5x - 3 = -7(x +1)\\-5(-2) - 3 = -7((-2) +1)\\10-3 =-7(-1)\\7=7

D. is correct.

Answer:

D. -5x-3=-7(x+1)

<em>Hope this was helpful.</em>

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