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# polynomial

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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

#1
+2293
+1

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
+2293
+1

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