Question

If α and β are zeros of polynomial 3x² -x+2 , then form a polynomial whose zeroes are 3α and 3β​

Answer 1

$\begin{gathered}\huge\blue{\mid{\fbox{\tt{SOLUTION}}\mid}} \\ \end{gathered}$

$Now. \: a \: + \beta = - \frac{b}{a} \\ = \frac{ - ( - 7)}{3}$

$\alpha \: + \beta = \frac{7}{3} ....(1) \\ and \: \alpha \beta = \frac{c}{a} = \frac{ - 6}{3}$

αβ=−2...(2)

(1) α2+β2=(α+β)2−2α+β

=(37)2−2(−2)     [ from (1) & (2)]$$

=949+4=949+36=985

α2β2=(αβ)2=(−2)2=4

 

The pohynomical whose roots are α2,β2 is given by , x2−(sumofroots)x+productofroots

=x2−985x+4

=99x2−85x+36

=91(9x2−85x+36)

(2) (2α+3β)+(3α+2β)=5α+5β

=5(α+β)

=5⋅37

=335

(2α+3β)+(3α+2β)=6α24αβ+9αβ+6β2

=6(α2+β2)+13αβ

=6{(α+β)2−2αβ}+Bαβ

=6(α+β)2−12αβ+Bαβ

=6(α+β)2+αβ

=6(37)2+(−2)

=6⋅949−2

=398−2

=398−6

=392

the required polynomial whose you are (2α+3β) and (3α+2β) is

x2−335x+392

=31(3x2−35x+92)

Answer 2

$\largex^{2}-x+6$

$\large\underline{\large\underline{\text{How I solved this problem}}}$

Firstly, I found the relation between the old and new roots. Then, I built a new polynomial $f(x)$.

As we know the relation of the pair of roots, we can find $g(x)$ then easily derive $f(g(g^{-1}))(x)=f(x).$

It is a composition of functions.

Why I introduced the method is that this method is conventional, even for higher degrees of polynomials.

$\large\underline{\large\underline{\text{Explanation}}}$

Let us assume new roots are,

$\cdots \longrightarrow \text{ x=\alpha' or x=\beta' .}$

The new and old roots satisfy,

$\cdots \longrightarrow \text{ \alpha '=3\alpha and \beta '=3\beta .}$

Let $f(x)$ be the required new polynomial. Then, since $\text{ x=\alpha ' or x=\beta ' }$ are new roots, it satisfies,

$\cdots \longrightarrow \text{ f( \alpha ')=0 and f( \beta ')=0 .}$

As we know,

$\cdots \longrightarrow \text{ f(x)=0 for x=3 \alpha and x=3 \beta .}$

So, the old polynomial is,

$\cdots \longrightarrow f(3x)=3x^{2}-x+2.$

$g(x)=3x$ then $g^{-1}(x)=\dfrac{1}{3}x$,

$\cdots \longrightarrow f(g(g^{-1}))(x)=\dfrac{1}{3} x^{2}- \dfrac{1}{3} x+2.$

As we know,

$\cdots \longrightarrow f(x)=\dfrac{1}{3} x^{2}- \dfrac{1}{3} x+2.$

Lastly, as we know we can multiply any nonzero number,

$\large\text{\underline{Answer}} \rightarrow \boxed{x^{2}-x+6.}$

$\large\underline{\large\underline{\text{Interesting facts}}}$

If a polynomial $f(x)=ax^{2}+bx+c$ has $\text{ x= \alpha or x=\beta }$ as roots, the new equation having reciprocal roots is $\text{ f(x)=cx^{2}+bx+a for ac \neq 0 .}$