Question

help me graduate please the graph of each function is shown . write the function in factored format. do not include complex numbers

Answer 1

Given the function of the graph:

`g(x)=2x^4+x^3-47x^2-25x-75`

Let's factor the given function.

Regroup the terms:

`g(x)=x^3-25x+2x^4-47x^2-75`
`\begin{gathered} \text{Factor x out of x}^3-25x\colon \\ \\ g(x)=x(x^2-25)+2x^4-47x^2-75 \end{gathered}`
`g(x)=x(x^2-5^2)+2x^4-47x^2-75`
`g(x)=x(x+5)(x-5)+2x^4-47x^2-75`
`\begin{gathered} \text{ Rewrite x}^4as(x^2)^2\colon \\ \\ g(x)=x(x+5)(x-5)+2(x^2)^2-47x^2-75 \end{gathered}`
`\begin{gathered} Letu=x^2 \\ \\ g(x)=x(x+5)(x-5)+2u^2-47u^{}-75 \end{gathered}`
`\begin{gathered} Factor\text{ by grouping:} \\ g(x)=x(x+5)(x-5)+(2u+3)(u-25) \end{gathered}`
`\begin{gathered} \text{ Repalce u with x}^2\colon \\ g(x)=x(x+5)(x-5)+(2x^2+3)(x^2-25) \end{gathered}`
`g(x)=x(x+5)(x-5)+(2x^2+3)(x^2-5^2)`
`g(x)=x(x+5)(x-5)+(2x^2+3)(x+5)(x-5)`
`\begin{gathered} \text{Factor out (x+5)(x-5)} \\ \\ g(x)=(x+5)(x-5)(x+2x^2+3) \\ \\ g(x)=(x+5)(x-5)(2x^2+x+3) \end{gathered}`

ANSWER:

`g(x)=(x+5)(x-5)(2x^2+x+3)`