**Precalculus 2.3a**

**I. Long Division of Polynomials:****A. Recall: When you did long division**

**Example 1:**

625/5 =

As you can see, 5 goes into 625 perfectly and there is no remainder so this means that 5 is a factor of 625.

**Example 2:**625/4 =

As you can see, 4 does not go into 625 perfectly, there is a remainder of 1 so we would write the answer like

156 + 1/4

You write the answer or quotient plus the remainder over the divisor

**B. Long Division of Polynomials:**Now try doing the same concept only using Polynomials:

**Example 3:**So therefore since (x - 4) is a factor so

x - 4 = 0

x = 4

Therefore (x - 4) is a factor and since there is not a remainder, then (x - h) is a factor and x = h.

**Example 4: **Therefore (x + 2) is a factor and since there is not a remainder, then (x - h) is a factor and x = h,

then x = -2

**Example 5:**Since when you divide by (x + 1), it has a remainder, it is not a factor and x = -1 is not a zero.

**Example 6:**So again, since (x - 2) is not a factor, x = 2 is not a zero.

**Precalculus 2.3b Synthetic Division:**To divide ax

^{3} + bx

^{2} + cx + d by (x - k), use the following pattern:

**Example 1: (5x**^{2} - 17x - 12) / (x - 4)Therefore, (5x

^{2} - 17x - 12) /(x - 4) = 5x + 3

since there is not a remainder!

**Example 2: (x**^{4} + 5x^{3} + 6x^{2} - x - 2) / (x + 2)

Again, since the remainder is 0,

(x^{4} + 5x^{3} + 6x^{2} - x - 2) / (x + 2) =

x^{3} + 3x^{2} - 1

**Example 3: (4x**^{3} - 13x + 10) / (x + 1)

Since there is not an x

^{2}, we put zero in that place:

Since 19 is the remainder:

(4x^{3} - 13 x + 10) / (x + 1) = 4x^{2} - 4x - 9 + 19/(x + 1)

**C. The Remainder Theorem: If a polynomial f (x) is divided by (x - k), the remainder is r = f (x)**

**Recall from Example 3, we saw the remainder = 19 so...**

f (-1) = 4(-1)^{3} - 13 (-1) + 10= 4 (-1) + 13 + 10= -4 + 13 + 10= 19

this gives the same answer!!

Let's try another:

**Example 4: (4x**^{3} - 13x + 10) / (x - 2) f(2) = 4(2)^{3} - 13(2) + 10 = 16, again, the same answer!

**D. The Factor Theorem:**

A polynomial f(x) has a factor (x - k) if and only if f(k) = 0.

**In Summary:**

**The remainder "r", obtained in the synthetic division of f (x) by (x - k) provides the following information:**

1. the remainder "*r*" gives the value of "*f*" at *x* = *k*. That is *r* = *f *(*k*)

2. If *r* = 0, then (*x* - *k*) is a factor of *f *(*x*)

3. If *r* = 0, then (*k*, 0) is an *x*-intercept of the graph of *f *(*x*).

**II. The Rational Zero Test:**

relates the possible rational zeros of a polynomial

Rational zero = p/q

where "p" and "q" have no common factors other than 1, "p" is a factor of the constant a_{0} and q is a factor of the leading coefficient a_{n}

**polynomial:**

f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{2}x^{2} + a_{1}x + a_{0}

Possible rational zeros = (factors of constant)/(factors of leading coefficient)

**Example 1:** f (x) = 4x^{5} - 8x^{4} - 5x^{3} + 10x^{2} + x - 2

So the possible rational zeros = (1, -1, 2, -2) / (1, -1, 2, -2, 4, -4) so

The possible zeros are : {1, -1, 1/2, -1/2, 1/4, -1/4, 2, -2}

Looking at the graph of the function, we see that the real zeros are:

(-1, 0), (-1/2, 0), (1/2, 0), (1, 0), and (2, 0)

**You can also use synthetic Division to eliminate possible zeros:**

f (x) = 4x^{5} - 8x^{4} - 5x^{3} + 10x^{2} + x - 2

try x = -1,

so f(x) = (x + 1)(4x

^{4} - 12x

^{3} + 7x

^{2} + 3x - 2)

so f (x) = (x - 1)(x + 1)(4x

^{3} - 8x

^{2} - 1x + 2)

Therefore f(x) = (x - 1)(x + 1)(x -2)(4x

^{2} - 1)

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

so x = -1, x = 1, x = 2, x = -1/2, x = 1/2

so you can see they are the same answers.

**III. Third Test for zeros:**

A real Number "b" is an upper bound for the real zeros of "f" if no zeros are greater than "b". Similarily, "b" is a lower bound if no real zeros of "f" are less than "b"

**Example 1:** f(x) = x^{4} - 4x^{3} + 15

1. degree is even

2. Leading coefficient is positive

so it rises to the left and rises to the right

3. points (-3, 204), (-2, 63), (-1, 20), (0, 15), (1, 12), (2, -1), (3, -12), (4, 15)

So from the points we chose, the lower bound will be 1 and the upper bound is 4

**We can use synthetic division to verify these thoughts:**

**Let x = -1, you can see that the remainder is 20 and the last row is:**

**1, -5, 5, -5, 20**

**Let x = 4, you can see that the remaider is 15 and the last row is:**

**1, 0, 0, 0, 15**

**Let x = 1, you can see that the remainder is 12 and the last row is:****1, -3, -3, -3, 12**

**Therefore we can conclude, given x = c:**

**1. If c is greater than zero and each number in the last row is either positive or zero, "c" is an upper bound for the real zeros of "f".**

**2. If c is less than zero and each number in the last row are alternately positive and negative, (zero counts as either positive or negative ), "c" is a lower bound for the real zeros of "f".**

**Therefore -1 is a lower bound and 4 is an upper bound. **