(a)

Remainder theorem : \ \

If a polynomial is divided by , the remainder is .

Factor theorem : \ \

A polynomial has a factor .

The polynomial function and factors are and .

By using remainder theorem and by using factor theorem is a factor of .

By using remainder theorem and by using factor theorem is a factor of .

(b)

By using remainder theorem and by using factor theorem is a factor of .

The resulting quadratic expression factorized as :

.

The complete factorization of is .

Thus, the remaining factor is .

(c)

The factors of are , , and .

The polynomial function is .

The complete factorization of is .

(d)

Consider .

To find the real zeros of the function, set equal to zero.

; ; . \ \

The real zeros of the function are , and .

(e)

The polynomial function is and factors are and .

The complete factorization of is .

The real zeros of the function are , and .

1. Draw a coordinate plane.

2. Plot the zeros.

3. Graph the curve.

Graph :

(a)

From remainder theorem, the factors are and .

(b)

The remaining factor is .

(c)

The complete factorization of is .

(d)

The real zeros of the function are , and .

(e)

Graph of the function is : \ \