Polynomial Factoring and Solving Equations Guide

End Behavior

Degree of a Polynomial

5-5 Factoring and Solving Polynomial Equations Pg. 342

Calculator Function – How to take the cube root of a number • To take the cube root of a number, press MATH, then select option 4. Example: What is ? 24

Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x3 – 2x2 = x4 – x3 – 3x2 + 3x = = = Grouping – common factor the first two terms and then the last two terms. Common Factor x2(x – 2) x(x3 – x2 – 3x + 3) x[x2(x – 1) – 3(x – 1)] Common Factor x(x2 – 3)(x – 1)

Solving Polynomial Equations The expressions on the previous slide are now equations: y = x3 – 2x2 and y = x4 – x3 – 3x2 +3x To solve these equations, we will be solving for x when y = 0.

Solve Let y = 0 y = x3 – 2x2 0 = x3 – 2x2 0 = x2(x – 2) x2 = 0 or x – 2 = 0 x = 0 x = 2 Therefore, the roots are 0 and 2. Common factor Separate the factors and set them equal to zero. Solve for x

Solve Let y = 0 y = x4 – x3 – 3x2 + 3x 0 = x4 – x3 – 3x2 + 3x 0 = x(x3 – x2 – 3x + 3) 0 =x[x2(x – 1) – 3(x – 1)] 0 = x(x – 1)(x2 – 3) x = 0 or x – 1 = 0 or x2 – 3 = 0 x = 0 x = 1 x = Therefore, the roots are 0, 1 and ±1.73 Common factor Group Separate the factors and set them equal to zero. Solve for x

The Quadratic Formula For equations in quadratic form: ax2 + bx + c = 0, we can use the quadratic formula to solve for the roots of the equation. This equation is normally used when factoring is not an option.

Remember, the root 0 came from an earlier step. Using the Quadratic Formula Solve the following cubic equation: y = x3 + 5x2 – 9x 0 = x(x2 + 5x – 9) x = 0 x2 + 5x – 9 = 0 We can, however, use the quadratic formula. Can this equation be factored? We still need to solve for x here. Can this equation be factored? YES it can – common factor. No. There are no two integers that will multiply to -9 and add to 5. a = 1 b = 5 c = -9 Therefore, the roots are 0, 6.41 and -1.41.

Factoring Sum or Difference of Cubes If you have a sum or difference of cubes such as a3 + b3 or a3 – b3, you can factor by using the following patterns. Note: The first and last term are cubed and these are binomials.

Example Factor x3 + 343. Note: This is a binomial. Are the first and last terms cubed? x3 + 343 = (x)3 + (7)3 = ( + )( - + ) x 7 x2 7x 49

Example Factor 64a4 – 27a = a(64a3 – 27) Note: Binomial. Is the first and last terms cubes? = a( (4a)3 – (3)3) Note: = a( - )( + + ) 4a 3 16a2 12a 9

FACTORING AND ROOTSCUBIC FACTORING Difference of Cubes a³ - b³ = (a - b)(a² + ab + b²) Sum of Cubes a³ + b³ = (a + b)(a² - ab + b²) Question: if we are solving for x, how many possible answers can we expect? 3 because it is a cubic!

Let’s try one • Factor a) x3-8 b) x3-125

Let’s Try One • 81x3-192=0 Hint: IS there a GCF???

Factor by Grouping Some four term polynomials can be factor by grouping. Example. Factor 3x3 + 7x2 +12x + 28 Step 1 Pair the terms. Step 2 Factor out common factor from each pair. Identical factors Step 3 Factor out common factor from each term.

Example Factor 3x3 + 7x2 -12x - 28 Step 1 Note: Subtraction is the same as adding a negative Step 2 Step 3 Note: This factor can be further factored

Solving Polynomial Equations Solve Set equation equal to zero. Factor. Set each factor equal to zero and solve.

Polynomial Factoring and Solving Equations Guide