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Name/Period: _______________________________________Unit icHomeworkHW √On-TimeMar. 49.19.2Adding & Subtracting PolynomialsMultiplying & FactoringPage 557: 3-26 oddPage 565: 3-35 oddEssential Question: How do we determine which numbers we can add together/subtract from one another?Mar. 89.3Multiplying BinomialsFactoringPage 572: 3, 5, 7, 11, 13, 15, 23-33 oddEssential Question: What property of numbers do we apply when multiplying binomials?Mar. 109.59.6Quiz #1FactoringPage 586: 3-17Page 596: 4-21Essential Question: How do you determine the factors of a quadratic equation?Mar. 149.79.8FactoringPage 603: 3-19Page 610: 3-11Essential Question: How do you determine the factors of a quadratic equation?Mar. 16Dividing PolynomialsPage 788: 3-18Essential Question: What do division and factoring have in common?Mar. 18Quiz #2Unit ReviewReview PacketStudy, study, studyEssential Question: How is factoring and simplifying radicals similar?Mar. 29Unit TestEssential Question: How do you determine the factors polynomial equations?VocabularyMonomial – a real number, a variable, or a product of a real number and one or more variables with whole number exponents. Examples of monomials:18z-4x22.5xy3a/3Degree of a monomial – the sum of the exponents of its variables.5xDegree is 1 (The exponent is 1).6x3y2Degree is 5 (The exponents are 3 and 2).4Degree is 0 (4 = 4x0; the degree of a nonzero constant is 0).Polynomial – a monomial or sum of monomials. For example, 3x4 + 5x2 – 7x + 1 is a polynomial.Standard form of a polynomial – the degrees of its monomial terms decrease from left to right.5x3 + 10x2 – 13x + 22 is written in standard form10x + 5x4 – 7x2 + 1 is not in standard formDegree of a polynomial – the degree of the monomial with the greatest exponent. For example, the degree of 3x4 + 5x2 – 7x + 1 is 4.Names of Polynomials:Monomial – polynomial with one term – 5x5Binomial-polynomial with two terms-5x + 9Trinomial-polynomial with three terms-4x2 + 7x + 3Adding & Subtracting PolynomialsYou add and subtract polynomials by combining like terms. You can use one of two methods:Method 1: Add vertically – Line up like terms. Then add the coefficients.Method 2: Add horizontally – Group like terms. Then add the coefficients.Examples. A researcher studied the number of overnight stays in the U.S. National Park Service campgrounds and in the backcountry of the national park system over a 5-year period. The researcher model the results, in thousands, with the following polynomials.Campgrounds:-7.1x2 – 180x + 5800Backcountry:21x2 – 140x + 1900In each polynomial, x = 0, corresponds to the first year in the 5-year period. What polynomial models the total number of overnight stays in both campgrounds and backcountry?Method 1Add vertically-7.1x2 – 180x + 5800+ 21x2 – 140x + 190013.9x2 – 320x + 7700Method 2Add horizontally(-7.1x2 – 180x + 5800) + (21x2 – 140x + 1900)(-7.1x2 + 21x2) + (-180x – 140x) + (5800 + 1900)13.9x2 – 320x + 7700A polynomial that models the number of stays (in thousands) in campgrounds and backcountry of the 5-year period is 13.9x2 – 320x + 7700.What is a simpler form of (x3 – 3x2 + 5x) – (7x3 + 5x2 – 12)?Method 1Subtract vertically x3 – 3x2 + 5x-(7x3 + 5x2 – 12)Need to distribute!!! x3 – 3x2 + 5x-7x3 – 5x2 + 12-6x3 – 8x2 + 5x +12Method 2Subtract horizontally(x3 – 3x2 + 5x) – (7x3 + 5x2 – 12)x3 – 3x2 + 5x – 7x3 - 5x2 + 12Need to distribute!!!(x3 – 7x3) + (-3x2 – 5x2) + 5x + 12-6x3 – 8x2 + 5x +12Practice. Simplify each sum or difference.(-12x3 + 106x2 – 241x +4477) + (14x2 – 14x +1545)(-4m3 – m + 9) – (4m2 + m -12)(x + 7x2) - (1 + 3x + x2)(7 + 2x – 4x2) + (-3x + x2 – 5)Multiplying & FactoringYou can use the Distributive Property to multiply a monomial by a polynomial.Examples.2x(3x + 1) = 2x(3x) + 2x(1)-x3(9x4 – 2x3 - 7)== 6x2 + 2x5n(3n3 – n2 + 8)=4x(3x2 – 2x + 6)=Practice.-3x(5x – 8)=(6n – 7)(5n3)=(7x2 + x – 3)(-2x2)=6x(2X3 + 7x)=4xy(2x + 7y)=x2y(3x2 – 9y)=Factoring a polynomial reverses the multiplication process. When factoring a monomial from a polynomial from a polynomial, the first step is to find the greatest common factor (GCF) of the polynomial’s terms.Find the GCF and then reverse distribute.What is the GCF of 5x3 + 25x2 + 45x?What is the GCF of 3x4 – 9x2 – 12x?Once you find the GCF, you can factor it out of the polynomial.Example. What is the factored form of 4x5 – 24x3 + 8x?Find the GCF.“Reverse distribute.”What is the factored form of 9x6 + 15x4 + 12x2?Using the Graphing Calculator to find Greatest Common FactorsPress the [MATH] keyArrow over to NUMArrow down (or press 9) to number “9: gcd(“ and press [ENTER]Enter “first number” “comma” “second number” then [ENTER] (Note: The numbers you are entering are the coefficients of the variables in the polynomial)The answer is the greatest common factor of the numbers.Practice. Factor each expression.14x4 – 21x22x2 + 812x2 + 27x-8x3 – 1633x5 – 121x230x4 + 45x3 – 105x18x3y + 24x2y2 – 42xy36x5 + 9x15m7n3 – 50m2n3210x6yz2 – 360x3yz4Multiplying BinomialsYou can use the distributive property to find the product of two binomials. Consider the first binomial as a single variable and distribute it to each term of the first binomial.Example. What is the simpler form of (3x – 7)(2x + 4)?Distribute the first factor, 3x – 7.(3x – 7)(2x + 4)=2x(3x – 7) + 4(3x – 7)Distribute 2xDistribute 4Combine like termsDistribution works well to multiply a binomial by a trinomial. What is the simpler form of (x + 3)(2x2 + x + 4)?Alternatively, you can use FOIL to multiply binomials. FOIL is an alternative to distribution when working with binomials.F-First-Multiply each of the first terms in the binomials.O-Outer-Multiply each of the outer terms in the binomials.I-Inner-Multiply each of the inner terms in the binomials.L-Last-Multiply each of the last terms in the binomials.Example. What is the simpler form of (5x – 3)(2x +1)?FirstOuterInnerLast(5x – 3)(2x +1)=(5x)(2x)+(5x)(1)+(-3)(2x)+(-3)(1)MultiplyCombine like termsWhat is the simpler form of (3x – 4)(x + 2)?Practice. What is the simpler form of each product? Use either distribution or FOIL.(3x – 4)(x + 2)(2x – 5)(3x2 + x – 1)(a + 4)(a2 + 3 – 2a) (x + 8)(x + 5)(x + 3)(x – 5)(x – 3)(x – 7)(2x – 7)(3x2 + x – 5)(2p2 + 3)(2p – 5)Factoring ax2 + bx + c, when a = 1Standard form for a quadratic equation is ax2 + bx + ca, b, and c are real numbers x is a variableTo factor, you find the two binomials that you could multiply to get the same quadratic equation.Factoring is “reverse-FOILing”. You end up with two binomials instead of starting with them.When a = 1, the factors will be (x +/- [real number 1])(x +/- [real number 2])The sum of the two real numbers is b.The product of the two real numbers is c.* real number 1 + real number 2 = b** real number 1 x real number 2 = c *Example. What is the factored form of x2 + 8x + 15?a = 1b = 8c = 15x2 + 8x + 15 = (x + 5)(x + 3)5 + 3 = 85 x 3 = 15Check using FOIL:(x)(x) + (3)(x) + (5)(x) + (3)(5)=x2 + 3x + 5x + 15=x2 + 8x + 15If c is positive, the 2 real numbers will have the same sign.If b is positive, the numbers will both be positive.(x + [real number 1])(x + [real number 2])If b is negative, the numbers will both be negative.(x - [real number 1])(x - [real number 2])Examples.b is POSITIVEb is NEGATIVEx2 + 3x + 2 = (x + 2)(x + 1)x2 - 5x + 6 = (x - 2)(x - 3)x2 + 7x + 12 = (x + 3)(x + ____)x2 - 8x + 12 = (x - 6)(x - ____)x2 + 9x + 20 = (x + ____)(x + ____)x2 - 10x + 16 = (x - _____)(x - ____)If c is negative, the 2 real numbers will have different signs.If b is positive, the number farthest from zero will be positive.If |real number 1| > |real number 2|(x + [real number 1])(x - [real number 2])If b is negative, the number farthest from zero will be negative.If |real number 1| > |real number 2|(x + [real number 1])(x - [real number 2])Examples.b is POSITIVEb is NEGATIVEx2 + x - 6 = (x + 3)(x - 2)x2 - 2x - 8 = (x + 2)(x - 4)x2 + 3x + 10 = (x + 5)(x - ____)x2 - 7x - 8 = (x + 1)(x - ____)x2 + 12x - 13 = (x + ____)(x - ____)x2 - 4x - 8 = (x + _____)(x - ____)a?cba?cba?cba?cba?cba?cbPractice. What is the factored form of each polynomial?r2 + 11r + 24x2 – 11x + 24n2 + 9n – 36c2 – 4c – 21 x2 – 2x – 35x2 + 6x – 55Factoring ax2 + bx = c when a > 1You can write some trinomials of the form ax2 + bx = c as the product of two binomials.Example. Factor the trinomial 6x2 + 23x + 7We are able to “break apart” and rewrite equations as long as the value remains constant. Therefore, we can rewrite the trinomial to be:Rewrite:6x2 + 2x + 21x + 723x = 2x + 21xFactor out GCF:2x(3x + 1) + 7(3x + 1)Distributive Property:(2x + 7)(3x + 1)Why can we do this? How does it work? What??????Reversing the Distributive Property – it’s like factoring.2n + 7n = n(2 + 7)5x + 10 = 5(x + 2) = 9n3a – 4ab = a()6xy + 2y =We can use an expression (usually a binomial) instead of a variable.4(x + 3) + x(x + 3) = (x + 3)(4 + x)(2n + 5)3 – (2n + 5)n = (2n+ 5)(3 – n)9(n- 3) – 2n(n – 3) =7(x – 2) + x(x – 2) = Factoring when a > 1Examples. What is the factored form of each trinomial?2x2 + 11x + 5a?cb3x2 + 5x + 2a?cb8x2 – 14x – 15a?cb2x2 + 21x – 11a?cba?cba?cbPractice.4x2 + 27x + 352x2 + 3x – 96x2 – 21x – 9 The area of a rectangle is 2x2 – 13x – 7. What are the possible dimensions of the rectangle? Use factoring.Factoring Special Cases Perfect-square Trinomials: The result of squaring a binomial.x2 + 8x + 16 = (x + 4)(x + 4)=(x + 4)24n2 – 12n + 9=(2n – 3)(2n – 3) =(2n – 3)2How to recognize a perfect-square trinomial.The first and last terms are perfect squares.Example: x2 + 8x + 16The first term, x2, is a perfect square of x.The last term, 16, is a perfect square of 4.The middle term is twice the product of one factor from the first term and one factor from the second term.Example: x2 + 8x + 16The middle term, 8x, is twice the product of x and 4.2?4?x=8xLet’s verify 16x2 – 56x + 49Example. What is the factored form of x2 -12x + 36?Practice. What is the factored form of each trinomial?x2 + 6x +94n2 – 12n + 9Factoring a Difference of Two SquaresFor all real numbers, a and b: a2 – b2 = (a + b)(a – b)Examples:x2 – 64 = (x + 8)(x – 8)25x2 – 36 = (5x + 6)(5x – 6)What is the factored form of x2 – 9?Practice. What is the factored form of each of the following expressions.v2 – 100 s2 – 1616x2 – 8125d2 – 64Factoring by GroupingYou can factor higher degree polynomials by using grouping.Example. What is the factored form of 3n3 – 12n2 + 2n – 8?Try grouping the two terms with the highest degrees together.(3n3 – 12n2) + (2n – 8)Factor out the GCF of each group of two terms.3n2(n – 4) + 2(n – 4)Factor out the common factor, n – 4(3n2 + 2)(n – 4)Check by FOIL.What is the factored form of 8t3 + 14t2 + 20t + 35?4q4 – 8q3 + 12q2 – 24qPractice. What is the factored form of 6h4 + 9h3 +12h2 +18h?Division of Polynomials & Prime PolynomialsDividing a polynomial by a monomial is very similar to factoring (finding the GCF).Examples.25x2+ 10x÷5x8x3-6x22x2=8x32x2-6x22x2=5x2+2=4x-33xy 6x2y+3xy-12xy2=2x+1-4yPractice. You can check your answers by multiplying the answer by the divisor (re-distribute)!24x4-16x3+8x2÷8x2x3-4xx7ab 42a2b+14ab-21ab218m5+45m4-36m39m3-33+44x-55x2÷(-11x)56m7n3-16m4n4-24m2n5-8mn3Prime Polynomials.A polynomial is prime if it cannot be factored.The standard form of a polynomial (quadratic equation) is ax2 + bx + c = 0The discriminant is b2 – 4ac.If b2 – 4ac is a positive number, it can be factored.If b2 – 4ac is a negative number, it cannot be factored.Prime!Are these prime?x2 + 6x – 72x2 + 6x – 5x2 – 4x + 43x2 – 4x – 6w2 – 6w + 16x2 – 11x + 24y2 + 19y + 60 ................
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