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Find number a and k so that x-2 is afactor OF F(x)=x4-2ax tax- X+k and F(-1)=3

Slav-nsk [51]

Question:

Find numbers a and k so that x-2 is a factor of

$f(x)=x^4-2ax^3+ax^2- x+k$

and

$f(-1)=3$

Answer:

$k = -2$ and $a=1$

Step-by-step explanation:

Given

$f(x)=x^4-2ax^3+ax^2- x+k$

$Factor:\ x - 4$

$f(-1)=3$

Required

Find a and k

For $f(-1)=3$

Substitute -1 for x

$f(x)=x^4-2ax^3+ax^2- x+k$

$f(-1) = (-1)^4 - 2a *(-1)^3 + a*(-1)^2 - (-1) + k$

$f(-1) = 1 - 2a *-1 + a*1 +1 + k$

$f(-1) = 1 +2a + a +1 + k$

$f(-1) = 2 +3a + k$

Substitute 3 for f(-1)

$3 = 2 +3a + k$

Collect Like Terms

$3 - 2 = 3a + k$

$1 = 3a + k$

Also:

If $x - 2$ is a factor, then

$f(2) = 0$

Substitute 2 for x and 0 for f(x)

$f(x)=x^4-2ax^3+ax^2- x+k$

$0 = 2^4 - 2a * 2^3 + a * 2^2 - 2 + k$

$0 = 16 - 2a * 8 + a * 4 - 2 + k$

$0 = 16 - 16a + 4a - 2 + k$

$0 = 16 - 12a - 2 + k$

Collect Like Terms

$2 - 16 = - 12a + k$

$-14 = k - 12a$

$k = 12a - 14$

Substitute 12a - 14 for k in $1 = 3a + k$

$1 = 3a + 12a - 14$

$1 = 15a - 14$

Collect Like Terms

$15a = 1 + 14$

$15a = 15$

Solve for a

$a = \frac{15}{15}$

$a=1$

Substitute 1 for a in $k = 12a - 14$

$k = 12 * 1 - 14$

$k = 12 - 14$

$k = -2$

3 0

2 years ago

The equation for the line of best fit for a data set is y = 2x + 1.5. If the point (1, 4) is in the data set, what is the residu

andriy [413]

It is related to econometrics, however using simple algebraic logic we can solve this problem. General formula with residual is $y=2x+1.5+r$ , where r is a residual. Plugging the pair into the formula, we obtain that $4=2*1+1.5+r$ and r=0.5

3 0

2 years ago

Read 2 more answers

Carisoprodol, a generic muscle relaxer, claims to have, on average, at least 120 milligrams of active ingredient. An independent

katovenus [111]

Answer:

The approximate p-value for the suitable test

0.05

|t| = |-1.5806| = 1.5806

t = 1.5806 < 2.009 at 0.05 level of significance

Carisoprodol, a generic muscle relaxer, claims to have, on average, is equal to 120 milligrams of active ingredient.

Step-by-step explanation:

Step(i):-

Given mean of the Population 'μ' = 120 milligrams

Given random sample size 'n' = 50

Given mean of the sample x⁻ = 116.2 milligrams

Standard deviation of the sample 'S' = 17 milligrams

Null hypothesis : 'μ' = 120

Alternative hypothesis : 'μ' < 120

Step(ii):-

Test statistic

$t = \frac{x^{-} -mean}{\frac{S}{\sqrt{n} } }$

$t = \frac{116.2 - 120}{\frac{17}{\sqrt{50} } } = \frac{-3.8}{2.404} = -1.5806$

Degrees of freedom

ν = n-1 = 50 -1 =49

t₀.₀₅ = 2.009

|t| = |-1.5806| = 1.5806

t = 1.5806 < 2.009 at 0.05 level of significance

Null hypothesis is accepted

Carisoprodol, a generic muscle relaxer, claims to have, on average, is equal to 120 milligrams of active ingredient.

P- value:-

The test statistic |t| = 1.5806 at 49 degrees of freedom

The test statistic value is lies between 0.05 to 0.1

0.05

3 0

2 years ago

Can you determine any purpose why these concepts are present in the pictures?

puteri [66]

The purpose why these concepts are present in the pictures is to develop imagination power among students.

As you imagine more, your mind becomes stronger and you have greater mental control. Innovation requires a strong imagination. If you succeed in what you set out to do, the same individuals who now accuse you of daydreaming will begin to aspire to be like you. Keep your aspirations high and your imagination expanding.

Imagination is a potent brain stimulant because it forces you to think differently than most people do. Your imagination can help you come up with scenarios that most effectively handle difficulties, even though your brain might classify them as being unexpected.

Learn more about imagination power here

brainly.com/question/8905110

#SPJ9

4 0

1 year ago

Anyone please help me!! It’s for a test!

PSYCHO15rus [73]

Yes because it’s the same shape just reflected and rotated

5 0

2 years ago