Factor and Solve Polynomial Equations Lesson 2.4

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Presentation transcript:

Factor and Solve Polynomial Equations Lesson 2.4 Honors Algebra 2 Factor and Solve Polynomial Equations Lesson 2.4

Goals Goal Rubric Factoring the sum and difference of two cubes. Factoring by grouping. Factoring in quadratic form. Factoring and solving polynomial equations. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Factored completely Factor by grouping Quadratic form

Factoring Polynomials Previously we learned how to factor the following types of quadratic expressions.

Factoring Polynomials We can also factor polynomials with degree greater than 2. Some of these polynomials can be factored completely using techniques learned earlier.

a. x3 + 2x2 – 15x = x(x2 + 2x – 15) = x(x + 5)(x – 3) b. 2y5 – 18y3 EXAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x3 + 2x2 – 15x = x(x2 + 2x – 15) Factor common monomial. = x(x + 5)(x – 3) Factor trinomial. b. 2y5 – 18y3 = 2y3(y2 – 9) Factor common monomial. = 2y3(y + 3)(y – 3) Difference of two squares c. 4z4 – 16z3 + 16z2 = 4z2(z2 – 4z + 4) Factor common monomial. = 4z2(z – 2)2 Perfect square trinomial

Factor the polynomial completely. Your Turn: for Examples 1 and 2 Factor the polynomial completely. 1. x3 – 7x2 + 10x x( x – 5)( x – 2) ANSWER 2. 3y5 – 75y3 3y3 (y – 5)( y + 5) ANSWER

Special Factoring Patterns Same as there is a special factoring pattern for the difference of two squares, there are also special factoring patterns that can be used to factor the sum or difference of two cubes.

Factor the polynomial completely. EXAMPLE 2 Factor the sum or difference of two cubes Factor the polynomial completely. a. x3 + 64 = x3 + 43 Sum of two cubes = (x + 4)(x2 – 4x + 16) b. 16z5 – 250z2 = 2z2(8z3 – 125) Factor common monomial. = 2z2 (2z)3 – 53 Difference of two cubes = 2z2(2z – 5)(4z2 + 10z + 25)

Factor the polynomial completely. Your Turn: for Examples 1 and 2 Factor the polynomial completely. 3. 16b5 + 686b2 2b2 (2b + 7)(4b2 –14b + 49 ) ANSWER 4. w3 – 27 (w – 3)(w2 + 3w + 9) ANSWER

Factor by Grouping Some polynomials can be factored by grouping pairs of terms that have a common monomial factor. The pattern for factoring by grouping is shown below:

Factor by Grouping Procedure Group the terms: If there are exactly 4 terms try: 2 & 2 grouping. If there are exactly 5 terms try: 3 & 2 grouping, or 2 & 3 grouping. Factor each group as if it were a factoring problem by itself. Now determine if the factored groups contain a common factor, if they do, factor it out and what is left makes up the other factor.

Factor the polynomial x3 – 3x2 – 16x + 48 completely. EXAMPLE 3 Factor by grouping Factor the polynomial x3 – 3x2 – 16x + 48 completely. x3 – 3x2 – 16x + 48 = x2(x – 3) – 16(x – 3) Factor by grouping. = (x2 – 16)(x – 3) Distributive property = (x + 4)(x – 4)(x – 3) Difference of two squares

Factoring in Quadratic Form An expression of the form 𝑎 𝑢 2 +𝑏𝑢+𝑐, where u is any expression in x, is said to be in quadratic form. example: 2 𝑥 6 +4 𝑥 3 −6=2 𝑥 3 2 +4 𝑥 3 −6 =2 𝑢 2 +4𝑢−6, where u = x3. The factoring techniques studied earlier to factor quadratic expressions can be used to factor expressions in quadratic form.

Factor completely: (a) 16x4 – 81 and (b) 2p8 + 10p5 + 12p2. EXAMPLE 4 Factor polynomials in quadratic form Factor completely: (a) 16x4 – 81 and (b) 2p8 + 10p5 + 12p2. a. 16x4 – 81 = (4x2)2 – 92 Write as difference of two squares. = (4x2 + 9)(4x2 – 9) Difference of two squares = (4x2 + 9)(2x + 3)(2x – 3) Difference of two squares Factor common monomial. b. 2p8 + 10p5 + 12p2 = 2p2(p6 + 5p3 + 6) Factor trinomial in quadratic form. = 2p2(p3 + 3)(p3 + 2)

Factor the polynomial completely. Your Turn: for Examples 3 and 4 Factor the polynomial completely. 5. x3 + 7x2 – 9x – 63 (x + 3)(x – 3) (x + 7) ANSWER 6. 16g4 – 625 (4g2 + 25)(2g + 5)(2g – 5) ANSWER 7. 4t6 – 20t4 + 24t2 4t2(t2 – 3)(t2 – 2 ) ANSWER

Solving Polynomial Equations by Factoring Earlier we learned how to use the zero product property to solve factorable quadratic equations. We can extend this technique to solve some higher-degree polynomial equations. First, write the polynomial equation in standard form (set equal to zero). Then, completely factor the polynomial expression and use the zero product property (set each factor equal to zero) to solve.

EXAMPLE 5 Standardized Test Practice SOLUTION 3x5 + 15x = 18x3 Write original equation. 3x5 – 18x3 + 15x = 0 Write in standard form. 3x(x4 – 6x2 + 5) = 0 Factor common monomial.

3x(x2 – 1)(x2 – 5) = 0 3x(x + 1)(x – 1)(x2 – 5) = 0 EXAMPLE 5 Standardized Test Practice 3x(x2 – 1)(x2 – 5) = 0 Factor trinomial. 3x(x + 1)(x – 1)(x2 – 5) = 0 Difference of two squares x = 0, x = – 1, x = 1, x = 5 ,or x = – 5 Zero product property The correct answer is D. ANSWER

Find the real-number solutions of the equation. Your Turn: for Example 5 Find the real-number solutions of the equation. 8. 4x5 – 40x3 + 36x = 0 ANSWER – 3, 3, – 1, 1, 0 9. 2x5 + 24x = 14x3 ANSWER – 3 , 3 , 2, 0, – 2 10. – 27x3 + 15x2 = – 6x4 9 + 41 4 ANSWER 0,

EXAMPLE 6 Solve a polynomial equation City Park You are designing a marble basin that will hold a fountain for a city park. The basin’s sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?

36 = (2x – 2)(x – 2)(x – 1) 0 = 2x3 – 8x2 + 10x – 40 EXAMPLE 6 Solve a polynomial equation SOLUTION 36 = (2x – 2)(x – 2)(x – 1) Write equation. 0 = 2x3 – 8x2 + 10x – 40 Write in standard form. 0 = 2x2(x – 4) + 10(x – 4) Factor by grouping. 0 = (2x2 + 10)(x – 4) Distributive property The only real solution is x = 4. The basin is 8 ft long, 4 ft wide, and 4 ft high. ANSWER

length: 12 ft; width: 6 ft; height: 2 ft Your Turn: for Example 6 11. WHAT IF? In Example 6, what should the basin’s dimensions be if it is to hold 40 cubic feet of water and have outer length 6x, width 3x, and height x? length: 12 ft; width: 6 ft; height: 2 ft ANSWER