Write a polynomial function f(x) (in factored form AND standard form) of least degree that has rational coefficients, a leading coefficient of 1, and -4, 1, and 2 as zeros.

Bit confused about the process. Any help would be appreciated!

To write a polynomial function f(x) x) (in factored form and standard form) of least degree that has rational coefficients, a leading coefficient of 1, and -4, 1, and 2 as zeros.

Solution :-

Given zeros are ,

[tex] \longrightarrow Zeros = -4 , 1 and 2 [/tex]

Since there are three zeroes , the polynomial will be a cubic polynomial .

We know that if , [tex] \alpha , \beta \ \& \ \gamma [/tex] are the zeros of the cubic polynomial then , we can write the polynomial as ,

[tex]\sf\longrightarrow f(x) =k ( x -\alpha)(x-\beta)(x-\gamma) [/tex]

where k is constant ,

Hence here ,

Hence here the polynomial can be written in factored form as ,

[tex]\sf\longrightarrow f(x) =1 [ x -(-4)] ( x-1)(x-2)\\[/tex]

[tex]\sf\longrightarrow \underset{\bf Factored\ form}{\underbrace{\underline{\underline{ f(x) = (x +4)(x-1)(x-2)}}}}[/tex]

Again we know that the Standard form of a cubic polynomial is ,

[tex]\sf\longrightarrow p(x) = ax^3+bx^2+ c x + d [/tex]

Now to find in standard form multiply the all three , as ,

[tex]\sf\longrightarrow f(x) = (x+4)(x-1)(x-2)\\ [/tex]

[tex]\sf\longrightarrow f(x) = [ x ( x-1) +4(x-1)] (x-2)\\ [/tex]

[tex]\sf\longrightarrow f(x) = [ x^2-x +4x -4](x-2)\\ [/tex]

[tex]\sf\longrightarrow f(x) = ( x^2+3x-4)(x-2)\\[/tex]

[tex]\sf\longrightarrow f(x) = x( x^2+3x-4)-2(x^2+3x-4) \\[/tex]

[tex]\sf\longrightarrow f(x) = x^3+3x^2-4x -2x^2-6x+8\\ [/tex]

Add like terms ,

[tex]\sf\longrightarrow \underset{\bf Standard\ form}{\underbrace{\underline{\underline{f(x) = x^3+x^2-10x+8}}}} [/tex]

Write a polynomial function f(x) (in factored form AND standard form) of least degree that has rational coefficients, a leading coefficient of 1, and -4, 1, and 2 as zeros.Bit confused about the process. Any help would be appreciated!