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The polynomial f(x) has degree 3.  If f(-1) = 15, f(0) = 0, f(1) = 0, and f(2) = 12, then what are the x-intercepts of the graph of f?

 Jun 30, 2022

Best Answer 

 #1
avatar+2293 
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Let the function be in the form \(f(x) = ax^3 + bx ^2 + cx + d \)

 

From \(f(0) = 0\), we know that \(d = 0\)

 

From the remaining 3, we can form the system: 

 

\(-a + b -c = 15\)                           (i)

\(a + b + c = 0\)                                 (ii)

\(8a + 4b + 2c = 12\)                        (iii)

 

Adding (i) and (ii) gives us \(2b = 15\), meaning \(b = 7.5\)

 

Substituting this into (iii) gives us: \(8a + 30 + 2c = 12\), which means \(8a + 2c = - 18\)

 

Substituting this into (ii) gives us: \(a + c = -7.5\)

 

Solving this new system, we find that \(a = -0.5\) and \(c = -7\)

 

This means that the polynomial is \(f(x)=-0.5x^3 + 7.5x^2 -7x \)

 

Now, recall the sum of the roots, by Vieta's, is \(-{b \over a} = 15\)

 

Because the polynomial is of degree 3, we know that there must be 3 real roots. 

 

But, recall that we know 2 of the roots, (0 and 1), so the remaining root is \(15 - 1 = 14\)

 

Thus, the x-intercepts are \(\color{brown}\boxed{0,1,14}\)

 Jun 30, 2022
 #1
avatar+2293 
+1
Best Answer

Let the function be in the form \(f(x) = ax^3 + bx ^2 + cx + d \)

 

From \(f(0) = 0\), we know that \(d = 0\)

 

From the remaining 3, we can form the system: 

 

\(-a + b -c = 15\)                           (i)

\(a + b + c = 0\)                                 (ii)

\(8a + 4b + 2c = 12\)                        (iii)

 

Adding (i) and (ii) gives us \(2b = 15\), meaning \(b = 7.5\)

 

Substituting this into (iii) gives us: \(8a + 30 + 2c = 12\), which means \(8a + 2c = - 18\)

 

Substituting this into (ii) gives us: \(a + c = -7.5\)

 

Solving this new system, we find that \(a = -0.5\) and \(c = -7\)

 

This means that the polynomial is \(f(x)=-0.5x^3 + 7.5x^2 -7x \)

 

Now, recall the sum of the roots, by Vieta's, is \(-{b \over a} = 15\)

 

Because the polynomial is of degree 3, we know that there must be 3 real roots. 

 

But, recall that we know 2 of the roots, (0 and 1), so the remaining root is \(15 - 1 = 14\)

 

Thus, the x-intercepts are \(\color{brown}\boxed{0,1,14}\)

BuilderBoi Jun 30, 2022

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