Math, asked by calebmontanez0, 1 month ago

Is (x-3) a factor of f(x)=x^4-4x^3-x^2+16x-12


Answers

Answered by llitzyourbfll
9

Answer:

Appropriate Question :-

If α and β are the zeroes of the quadratic polynomial f(x) = x² - px + q, then find the value of α² + β².

Given :-

α and β are the zeroes of the quadratic polynomial f(x) = x² - px + q.

To Find :-

What is the value of α² + β².

Solution :-

Given equation :

\dashrightarrow \sf\bold{x^2 - px + q}⇢x

2

−px+q

where,

a = 1

b = - p

c = q

Now, we have to find the sum and product of the zeroes :

\clubsuit♣ Sum of Zeroes :

\begin{gathered}\longmapsto \sf\boxed{\bold{\pink{Sum\: of\: Zeroes\: (\alpha + \beta) =\: \dfrac{- b}{a}}}}\\\end{gathered}

SumofZeroes(α+β)=

a

−b

Then,

\implies \sf \alpha + \beta =\: \dfrac{- (- p)}{1}⟹α+β=

1

−(−p)

\begin{gathered}\implies \sf \alpha + \beta =\: \dfrac{p}{1}\: \bigg\lgroup \bold{\purple{- \times - =\: +}} \bigg\rgroup\\\end{gathered}

⟹α+β=

1

p

−×−=+

\implies \sf\bold{\green{\alpha + \beta =\: p}}⟹α+β=p

Again,

\clubsuit♣ Product of Zeroes :

\begin{gathered}\longmapsto \sf\boxed{\bold{\pink{Product\: of\: Zeroes\: (\alpha\beta) =\: \dfrac{c}{a}}}}\\\end{gathered}

ProductofZeroes(αβ)=

a

c

Then,

\implies \sf \alpha\beta =\: \dfrac{q}{1}⟹αβ=

1

q

\implies \sf\bold{\green{\alpha\beta =\: q}}⟹αβ=q

Now, we have to find the value of α² + β² :

As we know that :

\longmapsto \sf\boxed{\bold{\pink{a^2 + b^2 =\: (a + b)^2 - 2ab}}}⟼

a

2

+b

2

=(a+b)

2

−2ab

We have :

α + β = p

αβ = q

According to the question by using the formula we get,

\leadsto \sf \alpha^2 + \beta^2 =\: (\alpha + \beta)^2 - 2\alpha\beta⇝α

2

2

=(α+β)

2

−2αβ

\leadsto \sf \alpha^2 + \beta^2 =\: (p)^2 - 2q⇝α

2

2

=(p)

2

−2q

\leadsto \sf\bold{\red{\alpha^2 + \beta^2 =\: p^2 - 2q}}⇝α

2

2

=p

2

−2q

\therefore∴ The value of α² + β² is p² - 2q.

Answered by xXCuteBoyXx01
78

Answer:

Step by Step Solution

STEP1:

Equation at the end of step 1

((((x4) - 22x3) - x2) + 16x) - 12

STEP2:

Polynomial Roots Calculator :

2.1 Find roots (zeroes) of : F(x) = x4-4x3-x2+16x-12

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

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