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Warm - up 6.2

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Factor:

1. 4x2 – 24x

4x(x – 6)

2. 2x2 + 11x – 21

(2x – 3)(x + 7)

3. 4x2 – 36x + 81

(2x – 9)2

Solve:

4. x2 + 10x + 25 = 0

X = -5

6.2 Polynomials and Linear Factors

Objective – To Analyze the factored form of a polynomial

CA State Standard

- 4.0 Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

- 10.0Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

Write the expression as a polynomial in standard form.

Example 1

(x + 1)(x + 1)(x + 2)

Multiply last two factors

(x + 1)(x2 + 2x + x + 2)

Combine 2x and x

(x+ 1)(x2 + 3x + 2)

Distribute x, then 1

x3 +3x2 + 2x

+ x2 + 3x + 2

Combine like terms

x3 +4x2 + 5x + 2

Write the expression in factored form.

Example 2

3x3 – 18x2 + 24x

3x(x2 – 6x + 8)

Factor out GCF 3x

3x(x – 2)(x – 4)

Factor trinomial using x-box

Relative Max – A point higher than all nearby points.

Relative Min – A point lower than all nearby points.

Relative Max

Relative Min

x-intercepts

Finding the zeros of a polynomial function in factored form

(use zero product property and set each linear factor equal

to zero)

Remember: the x-intercepts of a function are where y = 0, these values will now be referred to as the “zeros” of the polynomial.

Example 3

y = (x + 1)(x – 3)(x + 2)

x + 1 = 0

x – 3 = 0

x + 2 = 0

x = – 1

x = 3

x = – 2

You can write linear factors when you know the zeros. The relationship between the linear factors of a polynomial and the zeros of a polynomial is described by the Factor Theorem.

Factor Theorem:

The expression x – a is a linear factor of a polynomial if and only if the value “a” is a zero of the related polynomial function

Example 4

Write a polynomial function in standard form with zeros at -2, 3, 3.

-2 3 3

- The polynomial in the last example has three “zeros,” but it only has 2 distinct zeros: -2, 3.
- A repeated zero is called a multiple zero.
- A multiple zero has a multiplicity equal to the number of times the zero occurs.
(x – 2)(x +1)(x +1)

has 3 zeros: 2, -1, -1

since -1 is repeated, it has a multiplicity of 2

Example 5

Page 323 – 325

1, 8, 16, 29, and 30

Page 323-325 (1-11 odd, 17-35 odd, 65-72 all)