Question

question 5
0/1 pt 3 99 details
the polynomial of degree 4, p(x) has a root of multiplicity 2 at x = 3 and roots of multiplicity 1 at x = 0 and x = -3. it goes through the point (5,48).
find a formula for p(x).
p(x) =
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Explanation:

Step1: Write the general form of the polynomial

Since \(P(x)\) has a root of multiplicity 2 at \(x = 3\), a root of multiplicity 1 at \(x=0\) and \(x = - 3\), the general form of the polynomial is \(P(x)=a(x - 3)^{2}(x-0)(x + 3)\), which simplifies to \(P(x)=ax(x - 3)^{2}(x + 3)\).

Step2: Substitute the point \((5,48)\) into the polynomial

Substitute \(x = 5\) and \(P(5)=48\) into \(P(x)=ax(x - 3)^{2}(x + 3)\). We get \(48=a\times5\times(5 - 3)^{2}\times(5 + 3)\). First, calculate \((5 - 3)^{2}=4\) and \((5 + 3)=8\). Then the equation becomes \(48=a\times5\times4\times8\), that is \(48=a\times160\).

Step3: Solve for \(a\)

To find \(a\), we solve the equation \(48 = 160a\) for \(a\). So \(a=\frac{48}{160}=\frac{3}{10}\).

Step4: Write the final formula for \(P(x)\)

Substitute \(a=\frac{3}{10}\) back into \(P(x)=ax(x - 3)^{2}(x + 3)\), we have \(P(x)=\frac{3}{10}x(x - 3)^{2}(x + 3)\).

Answer:

\(P(x)=\frac{3}{10}x(x - 3)^{2}(x + 3)\)