5.4 Notes Synthetic Division, Remainder Theorem, and Factor Theorem

Bellwork: Read page 306. Use long division to divide 𝑥 3 −14 𝑥 2 +51𝑥−54 by 𝑥+2. Verify that you get the same answer as the book writers did when using synthetic division for the same problem. Homework: 5.4 Part 2 - 21-26, 29-30, 31-39 (odd), 41-44, 49-51, 71-73, 78, 83. Use the results of 49-51 to complete factor 𝑥 3 + 𝑥 2 −16𝑥−16. 5.4 Complete Solutions (in documents)

Long Division of Polynomials Subtract!! Subtract!! Subtract!!

Now let’s look at another method to divide… Why??? Sometimes it is easier…

Synthetic Division Synthetic Division is a ‘shortcut’ for polynomial division that only works when dividing by a linear factor (x + b). It involves the coefficients of the dividend, and the zero of the divisor.

(SUSPENSE IS BUILDING)

Example Divide: Step 1: Write the coefficients of the dividend in a upside-down division symbol. 1 5 6

Example Step 2: Take the zero of the divisor, and write it on the left. The divisor is x – 1, so the zero is 1. 1 1 5 6

Example Step 3: Carry down the first coefficient. 1 1 5 6 1

Example Step 4: Multiply the zero by this number. Write the product under the next coefficient. 1 1 5 6 1 1

Example Step 5: Add. 1 1 5 6 1 1 6

Example Step etc.: Repeat as necessary 1 1 5 6 1 6 1 6 12

Example The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original. 1 1 5 6 1 6 1 6 12

Synthetic Division The pattern for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.)

Synthetic Division This algorithm for synthetic division works only for divisors of the form x – k. Remember that x + k = x – (–k).

Using Synthetic Division Use synthetic division to divide x4 – 10x2 – 2x + 4 by x + 3. Solution: You should set up the array as follows. Note that a zero is included for the missing x3-term in the dividend.

Example – Solution cont’d Then, use the synthetic division pattern by adding terms in columns and multiplying the results by –3. So, you have .

Application of Long Division To begin, suppose you are given the graph of f (x) = 6x3 – 19x2 + 16x – 4.

Long Division of Polynomials Notice that a zero of f occurs at x = 2. Because x = 2 is a zero of f, you know that (x – 2) is a factor of f (x). This means that there exists a second-degree polynomial q (x) such that f (x) = (x – 2)  q(x). To find q(x), you can use long division.

Example - Long Division of Polynomials Divide 6x3 – 19x2 + 16x – 4 by x – 2, and use the result to factor the polynomial completely.

Example 1 – Solution Think Think Think Multiply: 6x2(x – 2). Subtract.

Example – Solution From this division, you can conclude that cont’d From this division, you can conclude that 6x3 – 19x2 + 16x – 4 = (x – 2)(6x2 – 7x + 2) and by factoring the quadratic 6x2 – 7x + 2, you have 6x3 – 19x2 + 16x – 4 = (x – 2)(2x – 1)(3x – 2).

Example 1 – Solution cont’d Note that this factorization agrees with the graph shown in Figure 2.28 in that the three x-intercepts occur at x = 2, x = , and x = . Figure 2.28

Long Division of Polynomials In Example 1, x – 2 is a factor of the polynomial 6x3 – 19x2 + 16x – 4, and the long division process produces a remainder of zero. Often, long division will produce a nonzero remainder. For instance, if you divide x2 + 3x + 5 by x + 1, you obtain the following.

Long Division of Polynomials In fractional form, you can write this result as follows. This implies that x2 + 3x + 5 = (x + 1)(x + 2) + 3 which illustrates the following theorem, called the Division Algorithm. Multiply each side by (x + 1).

Long Division of Polynomials

The Remainder and Factor Theorems

The Remainder and Factor Theorems The remainder obtained in the synthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. That is, to evaluate a polynomial function f (x) when x = k, divide f (x) by x – k. The remainder will be f (k).

Example Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x = –2. In other words, find 𝑓(−2). f (x) = 3x3 + 8x2 + 5x – 7 Solution: Using synthetic division, you obtain the following.

Example – Solution cont’d Because the remainder is r = –9, you can conclude that f (–2) = –9. This means that (–2, –9) is a point on the graph of f. You can check this by substituting x = –2 in the original function. Check: f (–2) = 3(–2)3 + 8(–2)2 + 5(–2) – 7 = 3(–8) + 8(4) – 10 – 7 = –9 r = f (k)

The Remainder and Factor Theorems Another important theorem is the Factor Theorem, stated below. This theorem states that you can test to see whether a polynomial has (x – k) as a factor by evaluating the polynomial at x = k. If the result is 0, (x – k) is a factor.

Example – Factoring a Polynomial: Repeated Division Show that (x – 2) and (x + 3) are factors of f (x) = 2x4 + 7x3 – 4x2 – 27x – 18. Then find the remaining factors of f (x). Solution: Using synthetic division with the factor (x – 2), you obtain the following. 0 remainder, so f (2) = 0 and (x – 2) is a factor of 𝑓.

Example – Solution 0 remainder, so f (–3) = 0 cont’d Take the result of this division and perform synthetic division again using the factor (x + 3). Because the resulting quadratic expression factors as 2x2 + 5x + 3 = (2x + 3)(x + 1) the complete factorization of f (x) is f (x) = (x – 2)(x + 3)(2x + 3)(x + 1). 0 remainder, so f (–3) = 0 and (x + 3) is a factor of 𝑓.

The Remainder and Factor Theorems For instance, if you find that x – k divides evenly into f (x) (with no remainder), try sketching the graph of f. You should find that (k, 0) is an x-intercept of the graph.

Can you… Use long division to divide polynomials by other polynomials. Use synthetic division to divide polynomials by binomials of the form (x – k). Use the Remainder Theorem and the Factor Theorem.

Questions???

Homework: 5.4 Part 2 - 21-26, 29-30, 31-39 (odd), 41-44, 49-51. Use the results of 49-51 to completely factor 𝑥 3 + 𝑥 2 −16𝑥−16. 71-73, 78, 83. This and preparing for your final is the only homework this week, so I expect 5.4 Part 2 to be 100% complete by next Monday. 5.4 Complete Solutions (in documents)