Question

write the standard - form equation for the graphed polynomial function with least degree. assume the root that touches the x - axis occurs twice.

• x^4 - 3x^3 + 3x^2 + 11x + 6

x^4 + 6x^3 - 11x^2 + 4x - 3
x^4 - 6x^3 + 11x^2 - 4x + 3
x^4 + 3x^3 - 3x^2 - 11x - 6

Explanation:

Step1: Recall polynomial - root relationship

If \(x = a\) is a root of a polynomial, then \((x - a)\) is a factor. The roots are \(x=-3\), \(x = - 1\), \(x = 2\), and since the graph touches the \(x\) - axis at \(x=-1\), the factor \((x + 1)\) has a multiplicity of 2.

Step2: Write the factored - form of the polynomial

The factored - form of the polynomial is \(y=a(x + 3)(x + 1)^2(x - 2)\). Let \(a = 1\) (since the leading - coefficient is not specified and we are looking for the polynomial with the least degree). First, expand \((x + 1)^2=x^{2}+2x + 1\).

Step3: Multiply \((x + 3)\) and \((x^{2}+2x + 1)\)

\((x + 3)(x^{2}+2x + 1)=x(x^{2}+2x + 1)+3(x^{2}+2x + 1)=x^{3}+2x^{2}+x+3x^{2}+6x + 3=x^{3}+5x^{2}+7x + 3\).

Step4: Multiply the result by \((x - 2)\)

\((x^{3}+5x^{2}+7x + 3)(x - 2)=x^{3}(x - 2)+5x^{2}(x - 2)+7x(x - 2)+3(x - 2)=x^{4}-2x^{3}+5x^{3}-10x^{2}+7x^{2}-14x + 3x-6=x^{4}+3x^{3}-3x^{2}-11x - 6\).

Answer:

\(x^{4}+3x^{3}-3x^{2}-11x - 6\)