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Chapter 10: Polynomials and Factoring!!! By Anastasia Stocker & Matthew Laredo Key Words • Monomial- a number and one or more variables with whole number exponents. • Degree of a monomial in one variable- the sum of the exponents of the variables in the monomial. • Polynomial-is a monomial or a sum of monomials (x’s with different degrees) • Binomial-a polynomial of two terms • Trinomial-polynomial of three terms • Standard form-the terms are arranged in decreasing order, from largest exponent to smallest exponent. • Degree of polynomial in one variable-the largest exponent of a variable. Monomials • Remember a monomial is just a number. It could or could not have an X or Y in it. • The “degree of the monomial” is simply the sum of the exponents. A.K.A. add the little numbers above the X’s and Y’s • Examples of monomials: 8, -2x, 3x²y or ½x² • Therefore the “degree of the monomials” would be 0, 1, 3 and 2…respectively. • The degree of 8 is zero because 8 = 8xº. • NOTE: Any term to the 0 power equals 1 i.e. xº=1 Section 10.1 POLYNOMIALS!!! Adding (5x³ + x² -2x + 7) + (3x² – 4x + 7) AND Subtracting (3x² – 5x + 3) – (2x² – x - 4) 5x³ + x² - 2x + 7 + 3x² – 4x + 7 5x³ + 4x² - 6x + 14 3x² – 5x + 3 – 2x² + x + 4 x² – 4x + 7 Adding & Subtracting Polynomials Remember when dealing with “polynomials”, aka numbers, group like terms. Put the x’s with the x’s and the x²’s and with x²’s and so on and so forth. ALSO!!!! Do not forget to distribute the subtraction sign within the parentheses. Oh, and these are parentheses ( ). Section 10.2 Multiplying Polynomials Using FOIL: Product of First terms + Product of Outer terms+ Product of Inner terms+ Product of Last terms (3x+4)(x+5)=3x²+15x+4x+20 Then combine like terms!! =3x²+19x+20. Using the Punnett Square • Take the terms inside of the parenthesis and put them on either side of the squares. • Then multiply to fill the boxes inside of the square. • For the final answer combine like terms (same colors) 3x + 4 X 3x² 4x + 6x 8 2 =3x² + 10x + 8 Multiply Binomials to Find an Area The window has a total height of 3x+10 and total width of 2x+6. The are of the window is represented by the product of the height and width. 5 3x • A= height ·width • A= (3x+10)(2x+6) • =6x²+18x+20x+60 • =6x²+38X+60 5 3 2x 3 Section 10.3 Special Products of Polynomials SUM and DIFFERENCE PATTERN (a+b)(a-b)= a² - b² for example: (3x-4)(3x+4)= 9x² - 16 SQUARE OF A BINOMIAL PATTERN EXAMPLE: (a+b)² = a² + 2ab + b² (a – b)(a +b)= a² + b² (a+b) ²= a² - 2ab + b² (5t-2)(5t + 2)= (5t) ² - 2² = 25t² - 4 To apply: 1st Write Pattern 2nd Apply pattern 3rd Simplify. USE THE SQUARE OF A BINOMIAL PATTERN Find the product. (3n + 4)² Write equation. (a + b) ² = a² + 2ab + b² Write pattern. (3n + 4)²= 3n² + 2(3n)(4) + 4² Apply the pattern using parenthesis. = 9n² + 24n + 16 Simplify (mulitply out). Finding the Area of Figure To find the area of a particular region within a figure 1st set the area of the region you want to FIND equal to 2nd the entire are of the figure. 3rd subtract the area of the region you do not want in total area. Section 10.4 Solving Quadratic Equations in Factored Form The Zero Product Property: Solving the equation (x-2)(x+3)=0 Set each factor to zero so that Solve for X x–2=0 & X=2 x+3=0 X = -3 A polynomial is in factored form when “it is written as a product of two or more factors”…. Basically when an equation has parentheses ( ), separating “terms”...things with the X’s or Y’s x² + 5x + 6 = 0 Not Factored (X + 2) (X + 3) = 0 FACTORED!!!!! ☻ A “repeated-factor” equation is when the equations factors are the same. Meaning the factors of (x+3)² are (x+3)(x+3). Therefore…when solving these equation, use the zero product property. (x + 5) ² = 0 X+5=0 X = -5 When solving a factored cubic equation, again…use the same method. i.e. (x+3)(x+4)(x+5)= 0 Graphing a factored equation??? NO PROBLEM!!! FIRST: Find the x-intercepts by solving the equation for X….like we just showed you how to. Second: Take the average of the x-intercepts to find the xcoordinate of the vertex Third: Plug the xcoordinate back into the original equation to find the vertex Fourth: Use the vertex and x-intercepts to sketch the graph y = (x + 2)(x - 3) X = -2 & 3 3 +(-2) 2 = ½ y = (½ + 2) (½ - 3) y = - 25 4 Figure it out, I cant hold your hand through everything Section 10.5 FACTORING x² + bx + c To factor a trinomial of this form means to write the product of the two binomials (factored form) x²+9x+14=(x+2)(x+7) TRINOMIAL BINOMIAL (FACTORED FORM) Remember: b & c are integers (numbers) How to Factor: When b and c are positive Factor x²+6x+8. What p and q could be: 1, 8 b=6 and c=8, so, therefore you need to find numbers where p times q is 8 and when added to together they equal 6. 2, 4 What p+q would equal: 1+8= 9, which does not = 6 2+4=6, since 6=6 you need these numbers! The Answer: x²+6x+8 = (x +2)(x +4) What about when there are negatives??? When negative numbers are added the same steps need to be taken to factor your equation. Just remember: p and q must multiply to equal c and add up to equal b Section 10.6 Factoring ax² + bx + c Factors of 6 Example: 6x² + 22x + 20= (3x+5)(2x+4) b=12+10= 22 Factors of 20 To factor ax² + bx + c is to find m and n whose product is a and numbers p and q whose product is c so that the middle term is the sum of the OUTER and INNER products of FOIL. mxn=a ax² + bx + c = (mx+p)(nx+q) pxq=c b= mq+np Example: factoring ax² + bx + c Factor 3x² + 5x – 2 For this trinomial, a=3, b=5, c= -2. Because c is negative, you need to find numbers p and q with different signs. • Write the numbers m and n whose product is 3 and the numbers p and q whose product is –x. • Use these numbers to write trial factors. Then use the Outer and Inner products of FOIL to check the middle term. m and n 1, 3 Trial factors (x-1)(3x+2) (x+1)(3x-2) (x+2)(3x-1) Answer- 3x² + 5x – 2= (x + 2)(3x – 1) p and q -1,2 1,-2 Middle term 2x-3x=-x -2x+3x=x -x+6x=5x Section 10.7 Factoring Special Products Difference of Two Squares Patterns EXAMPLE: a² - b² = (a - b) (a - b) 9x² - 25 = (3x + 5) (3x - 5) Perfect Square Trinomial Pattern a² + 2ab + b² = (a + b) ² EXAMPLE: x² + 14x + 49 = (x + 7) ² a² - 2ab + b² = (a - b) ² EXAMPLE: x² - 12x + 36 = (x - 6) ² Section 10.8 Factoring Cubic Polynomials When factoring a cubic polynomial, you should factor out the greatest common factor (GCF) first and then look for other patterns. Example: factor the greatest common factor out of 14x³-21x². 1st find the GCF of 14x³ and 21x². 14x³= 2· 7 · x · x · x 21x²= 3 ·7 ·x ·x GCF= 7 ·x ·x= 7x² Then use the distributive property to factor out the greatest common factor from each term. Answer- 14x³-21x²= 7x²(2x-3). Other ways to Factor Factor Completely Factor 4x³ + 20x² + 24x Solution: 4x³ + 20x² + 24x= 4x(x²+5x+6) Factor out GCF = 4x(x+2)(x+3) Factor trinomial Factor by Grouping Factor x³-2x²-9x+18 completely. Solution: x³-2x²-9x+18=(x³-2x²)+ (-9x+18) =x²(x-2)+(-9)(x-2)Factor each group =(x-2)(x²-9) Use distributive =(x-2)(x-3)(x+3) property. Factor difference of two squares. AND NOW YOU KNOW!!! THE END 23