Question

Write an expression in factored form for the polynomial of least possible degree graphed below.

y(x)= (blank)


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Answer 1

`\quad \huge \quad \quad \boxed{ \tt \:Answer }`

`\qquad\displaystyle \tt \rightarrow \: {y= -1/6(x⁴ +2x³-7x²-8x+ 12 }`

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`\large \tt Solution \: :`

The values of x for which curves cuts/touches the x - axis are the roots of that particular polynomial.

And that polynomial can be depicted in form :

`\qquad \tt \rightarrow \: a(x - h1) (x - h2) (x - h3)........ (x - hn) = 0`

[ where, h1, h2, h3... hn represents roots of that polynomial, and " a " is the stretch of curve]

And by that, we can sort out the roots of given polynomial that are :

• x = -3, -2, 1 and 2

Since there are four roots, the least degree polynomial formed will have bi - quadratic polynomial.

And it will be represented as :

`\qquad \tt \rightarrow \: y=a(x - ( - 3)) (x - ( - 2))(x - 1)(x - 2)`

`\qquad \tt \rightarrow \:y=a (x + 3)(x + 2)(x - 2)(x - 1)`

And it can be further solved to get ~

`\qquad \tt \rightarrow \: y=a(x + 3)( {x}^(2) - 4)(x - 1)`

`\qquad \tt \rightarrow \:y= a( {x}^(2) - 4)( {x}^(2) + 2x - 3)`

`\qquad \tt \rightarrow \: y=a( {x}^(4) + 2 {x}^(3) - 3 {x}^(2) - 4 {x }^(2) - 8x + 12)`

`\qquad \tt \rightarrow \: y=a({x}^(4) + 2 {x}^(3) - 7{x }^(2) - 8x + 12)`

Now, it's time to evaluate the value of a, for that we can just use a point that satifys the curve ( i.e (0 , -2)

plug in the values :

`\qquad\displaystyle \tt \rightarrow \: {-2= a(0⁴ +2(0)³-7(0)²-8(0) + 12 }`

`\qquad\displaystyle \tt \rightarrow \: {-2= a(0 +0-0-0 + 12 }`

`\qquad\displaystyle \tt \rightarrow \: {-2= a( 12 )}`

`\qquad\displaystyle \tt \rightarrow \: {a= -2 ÷ 12 }`

`\qquad\displaystyle \tt \rightarrow \: {a= -1/6}`

Therefore, the required equation is :

`\qquad\displaystyle \tt \rightarrow \: {y= -1/6(x⁴ +2x³-7x²-8x+ 12 }`

Answered by : ❝ AǫᴜᴀWɪᴢ ❞