DIVIDING POLYNOMIALS REMAINDER AND FACTOR THEOREMS FINDING ZEROS FOR POLYNOMIALS Section 2.5 – 2.7

Do-Now: Homework Quiz Factor the following expression. 12x 5 – 27x Solve the following equation. x 3 + 6x x = 0

5 th grade challenge

Polynomial Long Division Divide f(x) = 2x 4 + x 3 + x – 1 by x 2 + 2x – 1. Set your problem up like a long division problem. You must include a 0x 2 term.

Polynomial Long Division Divide f(x) = 3x x x – 11 by x + 3. Set your problem up like a long division problem.

Long division vs. synthetic division Find the value of the previous function when x = -3 using synthetic substitution. Why did I choose x = -3? What do you notice about the value of the function? What do you notice about the other numbers in the bottom row?

Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder is equal to the value of the function evaluated at x = k.

Additional example Divide using long and synthetic division. f(x) = 2x 3 + 9x x + 5 divided by x – 3.

Additional examples Perform the following division problems using synthetic substitution. 2x 4 – 8x 3 + 3x – 7 by x – 5  Make sure you put a zero in place of the missing term. x 3 + x 2 – 16x – 16 by x – 4. What is the remainder? What does this tell you about the number 4 and the expression x – 4?

Factor Theorem A polynomial f(x) has a factor x – k if and only if f(k) = 0.  This occurs when the remainder is 0 if f(x) is divided by x – k.  In other words, x – k is a factor because f(x) is evenly divisible by x – k.

Example

Factoring given a factor Factor the polynomial: 2x 3 – 11x 2 + 3x + 36 What method did you try. Why didn’t it work? What if I told you that one of the factors is x – 3? How would that allow you to factor the polynomial? Use synthetic division to divide by the given factor. Then see if you can factor the expression that results from the division.

Additional Example Factor x 3 + 9x x + 15 if x + 5 is one of the factors.

Using factoring to solve equations. Remember: If x – k is a factor of a polynomial, then  k is a zero of the polynomial  k is an x intercept of the graph of the polynomial  k is a root of the polynomial  if you substitute k into the equation the value of the equation is zero.  when the equation is set equal to zero, k is a solution. Example: If x – 4 is a factor, then 4 is a zero. Example: If x + 8 is a factor, then the graph has an x- intercept at -8. Example: If -5 is a zero of a polynomial, then x + 5 is a factor.

Example Solve the equation x 3 – 7x 2 + 2x + 40 = 0. One of the zeros is 5. Find the other two.

Additional Example Solve: 3x 3 + 4x 2 – 35x – 12 = 0. Hint: one of the solutions is x = 3.

EXAMPLE 6 Use a polynomial model BUSINESS The profit P (in millions of dollars ) for a shoe manufacturer can be modeled by P = – 21x x where x is the number of shoes produced (in millions). The company now produces 1 million shoes and makes a profit of $25,000,000, but would like to cut back production. What lesser number of shoes could the company produce and still make the same profit?

EXAMPLE 6 Use a polynomial model SOLUTION 25 = – 21x x Substitute 25 for P in P = – 21x x. 0 = 21x 3 – 46x + 25 Write in standard form. You know that x = 1 is one solution of the equation. This implies that x – 1 is a factor of 21x 3 – 46x Use synthetic division to find the other factors – – – 25 0

EXAMPLE 6 Use a polynomial model So, (x – 1)(21x x – 25) = 0. Use the quadratic formula to find that x ≈ 0.7 is the other positive solution. The company could still make the same profit producing about 700,000 shoes. ANSWER

Daily Homework Quiz ANSWER About 4300 or about One of the costs to print a novel can be modeled by C = x 3 – 10x x, where x is the number of novels printed in thousands. The company now prints 5000 novels at a cost of $15,000. What other numbers of novels would cost about the same amount?

HOMEWORK QUIZ Divide the polynomials using synthetic division. (2x 4 – x 3 + 4) ÷ (x + 1) Factor the following polynomial. (x – 10) is one of the factors. x 3 – 12x x + 80

Rational Zero Theorem

EXAMPLE 1 List possible rational zeros List the possible rational zeros of f using the rational zero theorem. a. f (x) = x 3 + 2x 2 – 11x + 12 Factors of the constant term: + 1, + 2, + 3, + 4, + 6, + 12 Factors of the leading coefficient: + 1 Possible rational zeros: +, +, +, +, +, Simplified list of possible zeros: + 1, + 2, + 3, + 4, + 6, + 12

EXAMPLE 1 List possible rational zeros b. f (x) = 4x 4 – x 3 – 3x 2 + 9x – 10 Factors of the constant term: + 1, + 2, + 5, + 10 Factors of the leading coefficient: + 1, + 2, + 4 +, +, +, +, + +, +, +, Possible rational zeros: Simplified list of possible zeros: + 1, + 2, + 5, + 10, +, +,

Finding all rational zeros Find all rational zeros for f(x) = x 3 – 8x x List all possible zeros. Then use synthetic division to guess and check.

Find all rational zeros. Find all rational zeros for f(x) = x 3 – 4x x + 18.

Find all REAL zeros For more difficult examples, use a graphing calculator to find reasonable estimates for rational zeros. Find all real zeros for f(x) = 10x 4 – 11x 3 – 42x 2 + 7x + 12

Find all REAL zeros Find all real zeros for f(x) = 2x 4 + 5x 3 – 18x 2 – 19x + 42

EXAMPLE 4 Solve a multi-step problem ICE SCULPTURES Some ice sculptures are made by filling a mold with water and then freezing it. You are making such an ice sculpture for a school dance. It is to be shaped like a pyramid with a height that is 1 foot greater than the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the dimensions of the mold?

Solve the following equation 3x 2 + 5x + 4 = 0 How many solutions are there? What kind of solutions are they?

Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, then the equation f(x) = 0 has exactly n solutions, provided that repeated solutions are counted multiple times. When is there a repeated solution?  When f(x) can be written with a factor raised to a power.  Ex: x 2 – 8x + 16 = 0……..(x – 4) 2 = 0 What does the graph look like when there is a repeated solution?  The graph touches the x-axis at that point, but does not cross it.

GUIDED PRACTICE for Example 2 Find all zeros of the polynomial function. 3. f (x) = x 3 + 7x x + 9 The zeros of f are – 1, −3, and – 3. ANSWER 4. f (x) = x 5 – 2x 4 + 8x 2 – 13x + 6 ANSWER Zeros of f are 1, 1, – 2, 1 + i 2, and 1 – i 2

EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and Using synthetic division, you can determine that – 1 is a zero repeated twice and 2 is also a zero. STEP 2 Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x 2 – 4x + 7. Therefore: f (x) = (x + 1) 2 (x – 2)(x 2 – 4x + 7) Find the zeros of a polynomial function

EXAMPLE 2 STEP 3 Find the complex zeros of f. Use the quadratic formula to factor the trinomial into linear factors. f(x) = (x + 1) 2 (x – 2) x – (2 + i 3 ) x – (2 – i 3 ) The zeros of f are – 1, – 1, 2, 2 + i 3, and 2 – i 3. ANSWER Find the zeros of a polynomial function