FCC MATH 8° 2P 3-7 FACTORING TRINOMIALS, DIFFERENCE OF SQUARE AND PERFECT CUBE BINOMIALS

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AREA: MATH SUBJECT: MATH GRADE EIGTH YEAR: 2020 PERIOD II CURRICULAR WEEK: 3-7 N° HOURS: AREA COMPONENT NUMERICAL TOPIC: FACTORING TRINOMIALS, DIFFERENCE OF SQUARE AND PERFECT CUBE BINOMIALS  Simplify or transform algebraic expressions by taking out single-term common factors. STANDARDS : LEARNING OBJECTIVE: To solve problems in different context by factoring trinomials in the form x2+bx+c ,ax2+bx+c and binomials in simple terms. LEARNING GOALS (Bloom’s Taxonomy) AREA COMPETENCIES Communcation Interpretar, Comparar, Seleccionar Describir

Reasoning Verificar, formular, utilizar Proponer, justificar, Clasificar, deducir

Resolution

INDICATORS BY LEVEL

To identify the general rules to factor algebraic expressions. To choose the best way to factor trinomials and binomials. To describe how to factor trinomials and binomials using different case of factoring. To use different case of factoring to factor trinomials and binomials. To classify expressions in those that can be factored using case of factoring for trinomials. To solve problems by factoring trinomials and binomials. To explain how to factor trinomials in different ways. To propose situations in which factoring can be used to solve real word problems. LEARNING-SPECIFIC VOCABULARY

Resolver, proponer, Utilizar, comprobar Determinar, explicar

 Algebraic expressions  Polynomials  Trinomials  Factor  Perfect square  Square root  Distributive property  Factoring by grouping.  Zero product property.  Prime polynomial LANGUAGE AND COMMUNICATION CONTENT Past tense verbs, transitional phrases, ordinal numbers LANGUAGE LEARNING

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LANGUAGE LEARNING

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CULTURAL MULTICULTURAL

COGNITIVE

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To factor a polynomial using distributive property you have to find the GCF of…. To identify a perfect square trinomial you have to identify perfect squares. INTEGRATION To apply factoring in problems involving vertical motion. Do a research about how scientist use algebra and factoring to find extraterrestrial life. EVALUATION CRITERIA (HOW)  Quiz about factoring trinomials in the form x2+bx+c and ax2+bx+c .  Quiz about difference of squares.

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 Quiz about perfect cube binomials.  Class activity about factoring trinomials and binomials.  Homework’s at home.  Good relationship with their partners.  Having a good behavior during the class.  Appreciate the importance of factoring algebraic expressions to represent variety of situations in our real life.

EXPLORATION CONTEXTUALIZATION

What is the difference between a trinomial and binomial?

According to the picture above answer the following questions: a) How can factoring be used to find the dimensions of a garden? b) Explain your reasoning.

THEORETICAL BASIS TRINOMIALS IN THE FORM X2+BX+C

When two numbers are multiplied, each number is a factor of the product. Similarly, when two binomials are multiplied, each binomial is a factor of the product. To factor some trinomials, you will use the pattern for multiplying two binomials. Study the following example.

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Observe the following pattern in this multiplication.

Notice that the coefficient of the middle term is the sum of m and n and the last term is the product of m and n. This pattern can be used to factor quadratic trinomials of the form x2+bx+c.

To determine m and n, find the factors of c and use a guess-and-check strategy to find which pair of factors has a sum of b. TRINOMIALS IN THE FORM AX2+BX+C For trinomials of the form x2+ bx +c, the coefficient of x2 is 1. To factor trinomials of this form, you find the factors of c whose sum is b. We can modify this approach to factor trinomials whose leading coefficient is not 1.

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Observe the following pattern in this product.

You can use this pattern and the method of factoring by grouping to factor 6x2 +17x -5. Find two numbers, m and n, whose product is 6*5 or 30 and whose sum is 17.

DIFFERENCE OF SQUARES Two terms that are squared and separated by a subtraction sign like this: a2 – b2 Useful because it can be factored into (a+b)(a−b)

PERFECT CUBE BINOMIAL

The other two special factoring formulas you'll need to memorize are very similar to one another; they're the formulas for factoring the sums and the differences of cubes. Here are the two formulas:

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To help with the memorization, first notice that the terms in each of the two factorization formulas are exactly the same. Then notice that each formula has only one "minus" sign. The distinction between the two formulas is in the location of that one "minus" sign: For the difference of cubes, the "minus" sign goes in the linear factor, a – b; for the sum of cubes, the "minus" sign goes in the quadratic factor, a2 – ab + b2. Some people use the mnemonic "SOAP" to help keep track of the signs; the letters stand for the linear factor having the "same" sign as the sign in the middle of the original expression, then the quadratic factor starting with the "opposite" sign from what was in the original expression, and finally the second sign inside the quadratic factor is "always positive". a3 ± b3 = (a [Same sign] b)(a2 [Opposite sign] ab [AlwaysPositive] b2) Whatever method best helps you keep these formulas straight, use it, because you should not assume that you'll be given these formulas on the test. You should expect to need to know them. Note: The quadratic portion of each cube formula does not factor, so don't waste time attempting to factor it. Yes, a2 – 2ab + b2 and a2+ 2ab + b2 factor, but that's because of the 2's on their middle terms. These sum- and difference-of-cubes formulas' quadratic terms do not have that "2", and thus cannot factor.

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MODELLING Example 1

Factor x2+6x+8

SOLUTION: In this trinomial, b =6 and c =8. You need to find two numbers whose sum is 6 and whose product is 8. Make an organized list of the factors of 8, and look for the pair of factors whose sum is 6.

Example 2.

Factor x2 -10x+16

SOLUTION In this trinomial, b=-10 and c =16. This means that m +n is negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 16, and look for the pair of factors whose sum is -10.

Example 3. Factor x2 +x-12

SOLUTION:

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In this trinomial, b=1 and c=-12. This means that m +n is positive and mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of -12, where one factor of each pair is negative. Look for the pair of factors whose sum is 1.

Example 4

Factor x2 -7x-18

SOLUTION: Since b=7 and c=18, m +n is negative and mn is negative. So either m or n is negative, but not both.

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Example 5 65 SOLUTION

Solve x2+5x=6. Check your solutions.

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Example 6 A sponsor for the school yearbook has asked that the length and width of a photo in their ad be 65 increased by the same amount in order to double the area of the photo. If the photo was originally 12 centimeters wide by 8 centimeters long, what should the new dimensions of the enlarged photo be?

SOLUTION

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Example 7 65 SOLUTION

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Factor 7x2 +22x+3

In this trinomial, a=7, b=22 and c= 3. You need to find two numbers whose sum is 22 and whose product is 7 • 3 or 21. Make an organized list of the factors of 21 and look for the pair of factors whose sum is 22.

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Example 8 65

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Factor 10x2 -43x+28

SOLUTION In this trinomial, a= 10, b=-43 and c=28. Since b is negative, m +n is negative. Since c is positive, mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 10*28 or 280, and look for the pair of factors whose sum is -43.

Example 9

Factor 3x2 +24x+45

SOLUTION: Notice that the GCF of the terms 3x2, 24x, and 45 is 3. When the GCF of the terms of a trinomial is an integer other than 1, you should first factor out this GCF.

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Example 10 SOLUTION:

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Solve 8a2-9a-5=4-3a. Check your solutions.

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Example 11

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Factor each binomial

a) n2-25 b) 36x2-49y2 SOLUTION: a)

b)

Example 12

Factor 48a3-12a

SOLUTION:

Example 13 SOLUTION:

Factor 2x4-162

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Factor 5x3+15x2-5x-15

SOLUTION:

Example 15

Solve 18x3=50x

SOLUTION:

Example 16

Factor x3-8

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SOLUTION This is equivalent to x3 – 23. With the "minus" sign in the middle, this is a difference of cubes. To do the factoring, I'll be plugging x and 2 into the difference-of-cubes formula. Doing so, I get:

x3 – 8 = x3 – 23 = (x – 2)(x2 + 2x + 22) = (x – 2)(x2 + 2x + 4)

Example 17

Factor 27x3+1

SOLUTION:

The first term contains the cube of 3 and the cube of x. But what about the second term? Before panicking about the lack of an apparent cube, I remember that 1 can be regarded as having been raised to any power I like, since 1 to any power is still just 1. In this case the power I'd like is 3, since this will give me a sum of cubes. This means that the expression they've given me can be expressed as:

(3x)3 + 13 So, to factor, I'll be plugging 3x and 1 into the sum-of-cubes formula. This gives me:

27x3 + 1 = (3x)3 + 13 = (3x + 1)((3x)2 – (3x)(1) + 12) = (3x + 1)(9x2 – 3x + 1)

Example 18

Factor x3y6-64

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SOLUTION: So I now know that, with the "minus" in the middle, this is a difference of two cubes; namely, this is:

(xy2)3 – 43 Plugging into the appropriate formula, I get:

x3y6 – 64 = (xy2)3 – 43 = (xy2 – 4)((xy2)2 + (xy2)(4) + 42) = (xy2 – 4)(x2y4 + 4xy2 + 16)

PRODUCTION CLASS ACTIVITY

1. Factor each trinomial

2. Find an expression for the perimeter of a rectangle with the given area. a) Area: x2+24x-81 b) Area: x2+13x-90

3. The triangle has an area of 40 square centimeters. Find the height h of the triangle

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4. Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime.

5. A rectangle with an area of 35 square inches is formed by cutting off strips of equal width from a rectangular piece of paper.

a) Find the width of each strip. b) Find the dimensions of the new rectangle.

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6.

7. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime.

8. The United States Coast Guard’s License Exam includes questions dealing with the breaking strength of a line. The basic breaking strength b in pounds for a natural fiber line is determined by the formula 900c2= b, where c is the circumference of the line in inches. What circumference of natural line would have 3600 pounds of breaking strength? 9. The width of a box is 9 inches more than its length. The height of the box is 1 inch less than its length. If the box has a volume of 72 cubic inches, what are the dimensions of the box?

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13. Factor in each case:

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INTEGRATION ACTIVITY -TRANSVERSALITY (complementary reading, research, projects) Use factoring to calculate how long will remain a baseball in the air. CULTURAL MULTICULTURAL

Do a research about how factoring are involved in the construction of missiles.

RESOURCES - Board - Computers - Notebook - Internet - Teachers BIBLIOGRAPHY  Algebra 1, Glencoe Mathematics, McGraw Hill , Unit 1 , Chapter 9. United States of America, 2002.  http://www.mathsisfun.com/definitions/difference-of-squares.html , Checked on 11 April 2017.  http://www.purplemath.com/modules/specfact2.htm , Checked on 11 April 2017.  https://cdn.kutasoftware.com/Worksheets/Alg2/Factoring%20A%20Sum%20and%20Difference%20of%20Cubes.p df , Checked on 11 April 2017.

Web pages in which you can play and learn with math:      

http://www.mat.ucm.es/deptos/am/guzman/juemat/juemat.htm http://www.mathsisfun.com/rational-numbers.html http://www.euler.us.es http://acertijos.net/21.htm www.matecreativa.com http://nti.educa.rcanaria.es/usr/matematicas/