Question

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!

Answer 1

To Find :-

• 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 ,

`\longrightarrow Zeros = -4 , 1 and 2`

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

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

`\sf\longrightarrow f(x) =k ( x -\alpha)(x-\beta)(x-\gamma)`

where k is constant ,

Hence here ,

• 
  `\alpha` = -4
• 
  `\beta` = 1
• 
  `\gamma` = 2
• k = 1

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

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

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

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

`\sf\longrightarrow p(x) = ax^3+bx^2+ c x + d`

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

`\sf\longrightarrow f(x) = (x+4)(x-1)(x-2)\\`

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

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

`\sf\longrightarrow f(x) = ( x^2+3x-4)(x-2)\\`

`\sf\longrightarrow f(x) = x( x^2+3x-4)-2(x^2+3x-4) \\`

`\sf\longrightarrow f(x) = x^3+3x^2-4x -2x^2-6x+8\\`

Add like terms ,

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

Answer 2

f(x) = (x+4)(x-1)(x-2)

Explanation:

We are given the roots as -4, 1, and 2

We are also given the leading coefficient as 1

We can write the polynomial as

f(x)= a( x- r1)(x-r2) (x-r3) ...... where rn are the roots and a is the leading coefficient

f(x)= 1( x- -4) (x-1)(x-2)

= 1(x+4) (x-1) (x-2)

We can drop the leading 1 since it is not needed

f(x) = (x+4)(x-1)(x-2)