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# Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.

**Solution:**

Let us consider a fourth degree polynomial

f(x) = x^{4} - x^{3} - x^{2} - x - 2

From the remainder theorem,

f(-1) = 1 + 1 - 1 + 1 - 2 = 0

So (x + 1) is a factor.

f(2) = 16 - 8 - 4 - 2 - 2 = 0

So (x - 2) is a factor.

(x + 1)(x - 2) = x^{2} - 2x + x - 2

= x^{2} - x - 2

Use long division method to find the remaining factors

f(x) = (x + 1)(x - 2)(x^{2} + 1)

We know that (x^{2} + 1) contains no real factors

x^{2} + 1 = (x + i)(x - i)

It has a pair of conjugate zeros +i and - i.

Therefore, f(x) = x^{4} - x^{3} - x^{2} - x - 2 is an example and has fewer x-coordinates.

## Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.

**Summary:**

f(x) = x^{4} - x^{3} - x^{2} - x - 2 is an example and has fewer x-intercepts than its number of roots.

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