6-2 Polynomials & Linear Factors M11.D.2.2.2: Factor Expressions, including differences of squares and trinomials

Objectives The Factored Form of a Polynomial Factors & Zeros of a Polynomial Function

Writing a Polynomial in Standard Form Write (x – 1)(x + 3)(x + 4) as a polynomial in standard form. (x – 1)(x + 3)(x + 4) = (x – 1)(x2 + 4x + 3x + 12) Multiply (x + 3) and (x + 4). = (x – 1)(x2 + 7x + 12) Simplify. = x(x2 + 7x + 12) – 1(x2 + 7x + 12) Distributive Property = x3 + 7x2 + 12x – x2 – 7x – 12 Multiply. = x3 + 6x2 + 5x – 12 Simplify. The expression (x – 1)(x + 3)(x + 4) is the factored form of x3 + 6x2 + 5x – 12.

Writing a Polynomial in Factored Form Write 3x3 – 18x2 + 24x in factored form. 3x3 – 18x2 + 24x = 3x(x2 – 6x + 8) Factor out the GCF, 3x. = 3x(x – 4)(x – 2) Factor x2 – 6x + 8. Check: 3x(x – 4)(x – 2) = 3x(x2 – 6x + 8) Multiply (x – 4)(x – 2). = 3x3 – 18x2 + 24x Distributive Property

Vocabulary A relative maximum is the greatest y-value of the points in a region of the graph. A relative minimum is the smallest y-value of the points in a region of the graph. Relative Maximum Relative Minimum

Finding Relative Minimum and Maximum Values Using a graphing calculator, find the relative maximum, relative minimum, and zeros of each function. x³ + 6x² + 9x + 1 Step 1: Go to y= and enter the function and then graph Step 2: Hit 2nd Calc : Maximum Left Bound? : Move your cursor to the left of the “peak” and hit enter Right Bound?: Move your cursor to the right of the “peak” and hit enter Guess? : Just hit enter again Step 3: Hit 2nd Calc : Minimum Follow the same steps as 2 Step 4: Hit 2nd Calc : Zero Follow the same steps as 2, but your cursor needs to be to the left of where the function intersects the x-axis.

Finding Zeros of a Polynomial Function Find the zeros of y = (x + 1)(x – 1)(x + 3). Then graph the function using a graphing calculator. Using the Zero Product Property, find a zero for each linear factor. x + 1 = 0    or     x – 1 = 0    or    x + 3 = 0 x = –1 x = 1 x = –3 The zeros of the function are –1, 1, –3. Now sketch and label the function.

Writing a Polynomial Function From its Zeros Write a polynomial in standard form with zeros at 2, –3, and 0. 2 –3 0 Zeros ƒ(x) = (x – 2)(x + 3)(x) Write a linear factor for each zero. = (x – 2)(x2 + 3x) Multiply (x + 3)(x). = x(x2 + 3x) – 2(x2 + 3x) Distributive Property = x3 + 3x2 – 2x2 – 6x Multiply. = x3 + x2 – 6x Simplify. The function ƒ(x) = x3 + x2 – 6x has zeros at 2, –3, and 0.

In the example above, the multiplicity is 2 Vocabulary Let’s look at an example: has zeros at -2, 3, & 3. If a linear factor of a polynomial is repeated, it is called a multiple zero. A multiple zero has a multiplicity equal to the number of times a zero occurs. In the example above, the multiplicity is 2

Finding the Multiplicity of a Zero Find any multiple zeros of ƒ(x) = x5 – 6x4 + 9x3 and state the multiplicity. ƒ(x) = x5 – 6x4 + 9x3 ƒ(x) = x3(x2 – 6x + 9) Factor out the GCF, x3. ƒ(x) = x3(x – 3)(x – 3) Factor x2 – 6x + 9. Since you can rewrite x3 as (x – 0)(x – 0)(x – 0), or (x – 0)3, the number 0 is a multiple zero of the function, with multiplicity 3. Since you can rewrite (x – 3)(x – 3) as (x – 3)2, the number 3 is a multiple zero of the function with multiplicity 2.

Homework Pg 317 # 1, 2, 8, 10, 13, 21, 22, 29