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Suppose f(x) is a degree three polynomial with ⋅ a root of multiplicity 2 at x=1⋅ a root of multiplicity 1 at x=−2⋅ a y-intercept at (0,4).Find a formula for f(x).

Sagot :

Since x=1 is a root of multiplicity 2 and x= -2 is a root with multiplicity 1, our polynomial has the form

[tex]f(x)=A\cdot(x-1)^2\cdot(x+2)[/tex]

where A is an unknown constant.

We can find A by substituting into our last result the given point (0,4), that is,

[tex]4=A\cdot(0-1)^2\cdot(0+2)[/tex]

which gives

[tex]\begin{gathered} 4=A\cdot(-1)^2\cdot2 \\ \text{then} \\ 4=A\cdot1\cdot2 \\ so \\ 4=2A \end{gathered}[/tex]

then, by dividing both sides by 2, we get

[tex]A=2[/tex]

Therefore, the answer is:

[tex]f(x)=2(x-1)^2(x+2)[/tex]

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