holt mcdougal algebra 2 factoring polynomials use the factor theorem to determine factors of a...

Holt McDougal Algebra 2

Factoring Polynomials

Use the Factor Theorem to determine factors of a polynomial.Factor the sum and difference of two cubes.

Objectives



Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0. The Remainder Theorem states that if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.



Determine whether the given binomial is a factor of the polynomial P(x).

Example 1: Determining Whether a Linear Binomial is a Factor

A. (x + 1); (x2 – 3x + 1) B. (x + 2); (3x4 + 6x3 – 5x – 10)



Check It Out! Example 1

Determine whether the given binomial is a factor of the polynomial P(x).

a. (x + 2); (4x2 – 2x + 5) b. (3x – 6); (3x4 – 6x3 + 6x2 + 3x – 30)



Zero-Product Property

Example 1:




Example 2:

-10m3 – 2m2




Example 3:



Factoring Trinomials

Example 1:



Factoring trinomials Example 2:




3x2 – 5x + 2


Factoring PolynomialsDetermine whether the given polynomial is a factor of the polynomial P(x):

1.(x – 2); P(x) = 5x3 + x2 – 72.(x – 8); P(x) = x5 – 8x4 + 8x – 64Use the zero-product property to solve the equation:3.(t + 4)(t – 4) = 04.(c + 7)(c + 2) = 05.( d + ¾)(d + 5/8) = 06.(x + 5.4)(x – 3) = 07.(5y – 1)(2y + 4) = 08.(7t + 21)(t + 9) = 09.(2n – 7)(5n + 20) = 010. (2x – ½)(2x + ½) = 0




6x3 + 24x2 + 24x




2t3 – 2t2 – 12t




7x2 – 12x + 5



Factor: x3 – x2 – 25x + 25.

Example 2: Factoring by Grouping



Example 2 Continued

Check Use the table feature of your calculator to compare the original expression and the factored form.

The table shows that the original function and the factored form have the same function values.



Check It Out! Example 2a

Factor: x3 – 2x2 – 9x + 18.



Check It Out! Example 2a Continued

Check Use the table feature of your calculator to compare the original expression and the factored form.

The table shows that the original function and the factored form have the same function values.



Check It Out! Example 2b

Factor: 2x3 + x2 + 8x + 4.



Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring the sum or difference of two cubes.



Example 3A: Factoring the Sum or Difference of Two Cubes

Factor the expression.

4x4 + 108x



Example 3B: Factoring the Sum or Difference of Two Cubes


125d3 – 8



Check It Out! Example 3a


8 + z6



Check It Out! Example 3b


2x5 – 16x2



Example 4: Geometry ApplicationThe volume of a plastic storage box is modeled by the function V(x) = x3 + 6x2 + 3x – 10. Identify the values of x for which V(x) = 0, then use the graph to factor V(x).



Check It Out! Example 4

The volume of a rectangular prism is modeled by the function V(x) = x3 – 8x2 + 19x – 12, which is graphed below. Identify the values of x for which V(x) = 0, then use the graph to factor V(x).



4. x3 + 3x2 – 28x – 60

Lesson Quiz

2. x + 2; P(x) = x3 + 2x2 – x – 2

1. x – 1; P(x) = 3x2 – 2x + 5

3. x3 + 3x2 – 9x – 27

4. 64p3 – 8q3

holt mcdougal algebra 2 factoring polynomials use the factor theorem to determine factors of a...

Documents