Write each polynomial in irreducible factored form over
the set of real numbers and as the product of

linear factors over the set of complex numbers. Identify which zeros correspond
to x-intercepts and

which do not correspond to x-intercepts.

**1. f(x) = x ^{2} – 1
**= (x–1)(x+1) is both Irreducible Factored Form and Complex Linear Factored Form

x = 1 and x = –1 are zeros and x intercepts

**2. f(x) = x ^{2} + 1** Irreducible Factored Form

= (x– i)(x+ i) Complex Linear Factored Form

x = i and x = – i are zeros but not intercepts. f(x) has does not have any x intercepts

is both Irreducible Factored Form and Complex Linear Factored Form

and are zeros and x intercepts

Complex Linear Factored Form

and are zeros but not intercepts; f(x) has does not have any x intercepts

= x

= x

= x

= x

x = 0, 5, –5, i, – i are zeros; x = 0, 5, –5, are x-intercepts

The graph crosses the x-axis at x = 5 and x = –5

x = 0 is a zero of even multiplicity 2, so the graph touches but does not cross the x-axis at x = 0

x

= x

= (x

both Irreducible Factored Form

and Complex Linear Factored Form

, are zeros and x intercepts

= x

= (x

Complex Linear Factored Form

are zeros ; x = 3 is the only x intercept

**8. f(x) = x ^{2} + 2x + 3** Irreducible Factored Form

Irreducible Quadratic x

find complex zeros and use the complex zeros to create the factors.

Complex Linear Factored
Form

and are zeros but not intercepts; f(x) has does not have any
x intercepts

**9. f(x) = x ^{3} + 64** sum of two cubes

= (x+4)(x

Complex Linear
Factored Form

and are zeros ; x = –4 is the only x intercept

**10. f(x) = x ^{3} – 64** difference of two cubes

= (x–4)(x

Complex Linear
Factored Form

and are zeros: x = 4 is the only x intercept

**11. f(x) = x ^{9} – 4x^{8} + 5x^{7}**

= x

x

(x

Use quadratic formula to find complex zeros and use them to create the factors.

f(x) = x^{7} (x –(2+i)) (x–(2–i)) Complex Linear Factored
Form

x = 0, x = 2+i and x = 2–i are zeros

x^{7} is a repeated linear factor (x–0)^{7}

x = 0 is the only x intercept, of multiplicity 7;

The graph crosses the x axis once, at (0,0) and it flattens out as it crosses

because x = 0 is a zero of odd multiplicity greater than 1

The answer to number 12 was printed on the back side of
your handout for Fundamental Theorem of

Algebra, so is not posted here.