Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying Monomials Dividing by Monomials Multiplying Polynomials Mulitplying a Polynomial by a Monomial Special Binomial Products Solving Equations by Factoring Definitions of Monomials, Polynomials and Degrees Return to Table of Contents A monomial is a one-term expression formed by a number, a variable, or the product of numbers and variables. Examples of monomials y4z 17x2 4x 28 mn3 rt 6 32,457 a + b -5 5x + 7 x2(5 + 7y) 6+5rs 7x3y5 - 4 Monomials Drag the following terms into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term will disappear. 48x2yz3 4(5a2bc2) t xy4 7 A polynomial is an expression that contains two or more monomials. Examples of polynomials a2 8x3+x2 c2+d 8a3-2b2 4c-mn3 rt 6 a4b b+c2+4d3 Degrees of Monomials The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant such as 5 or 12 is 0. The constant 0 has no degree. Examples: 1) The degree of 3x is 1. The variable x has a degree 1. 2) The degree of -6x3y is 4. The x has a power of 3 and the y has a power of 1, so the degree is 3+1 =4. 3) The degree of 9 is 0. A constant has a degree 0, because there is no variable. 1What is the degree of ? A 0 B 1 C 2 D 3 2 A 0 B 1 C 2 D 3 3What is the degree of 3 ? A 0 B 1 C 2 D 3 4What is the degree of ? Degrees of Polynomials Example: Find degree of the polynomial 4x3y2 - 6xy2 + xy. The monomial 4x3y2 has a degree of 5, the monomial 6xy2 has a degree of 3, and the monomial xy has a degree of 2. The highest degree is 5, so the degree of the polynomial is 5. The degree of a polynomial is the same as that of the term with the greatest degree. Find the degree of each polynomial 1) 3 2) 12c3 3) ab 4) 8s4t 5) 2 - 7n 6) h4 - 8t 7) s3 + 2v2y2 - 1 Answers: 1) 0 2) 3 3) 2 4) 5 5) 1 6) 4 7) 4 5What is the degree of the following polynomial: A 3 B 4 C 5 D 6 6 A 3 B 4 C 5 D 6 7 A 3 B 4 C 5 D 6 8 A 3 B 4 C 5 D 6 A dding and Subtracting Polynomials Return to Table of Contents Standard Form Th e standard form of an equation is to put the terms in order from highest degree to the lowest degree. Standard form is commonly excepted way to write polynomials. Example: is in standard form. Put the following equation into standard form: Monomials with the same variables and the same power are like terms. Like TermsUnlike Terms 4x and -12x -3b and 3a x3y and 4x3y 6a2b and -2ab2 Combine these like terms using the indicated operation. 9Simplify A B C D 10Simplify A B C D 11Simplify A B C D To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (3x2 +5x -12) + (5x2 -7x +3) (3x4 -5x) + (7x4 +5x2 -14x) line up the like terms line up the like terms 3x2 + 5x x4 -5x (+)5x2 - 7x + 3 (+) 7x4 +5x2 - 14x 8x2 - 2x x4 +5x2 - 19x = We can also add polynomials horizontally. (3x2 + 12x - 5) + (5x2 - 7x - 9) Use the communitive and associative properties to group like terms. (3x2 + 5x2) + (12x + -7x) + ( ) 8x2 + 5x - 14 12Add A B C D 13Add A B C D 14Add A B C D 15Add A B C D 16Add A B C D To subtract polynomials, subtract the coefficients of like terms. Example: -3x - 4x = -7x 13y - (-9y) = 22y 6xy - 13xy = -7xy We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. Distribute the -1 and then follow the rules for adding polynomials (3x2 +4x -5) - (5x2 -6x +3) (3x2+4x-5) +(-1) (5x2-6x+3) (3x2+4x-5) + (-5x2+6x-3) 3x2 + 4x - 5 (+) -5x2 - 6x x2 +10x - 8 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. Distribute the -1 and then follow the rules for adding polynomials (4x3 -3x -5) - (2x3 +4x2 -7) (4x3 -3x -5) +(-1)(2x3 +4x2 -7) (4x3 -3x -5) + (-2x3 -4x2 +7) 4x3 - 3x - 5 (+) -2x3 - 4x x3 - 4x2 - 3x + 2 We can also subtract polynomials horizontally. (3x2 + 12x - 5) - (5x2 - 7x - 9) Change the subtraction to adding a negative one and distribute the negative one. (3x2 + 12x - 5) +(-1)(5x2 - 7x - 9) (3x2 + 12x - 5) + (-5x2 + 7x + 9) Use the communitive and associative properties to group like terms. (3x2 +-5x2) + (12x +7x) + (-5 +9) -2x2 + 19x + 4 17Subtract A B C D 18Subtract A B C D 19Subtract A B C D 20Subtract A B C D 21Subtract A B C D 22 What is the perimeter of the following figure? (answers are in units) A B C D M ultiplying Monomials Return to Table of Contents Recall: a monomial is a one-term expression formed by a number, a variable, or the product of numbers and variables. Now that we know how to add and subtract polynomials, we need to learn how to multiply them. Let's start with two monomials. Write this expression in expanded form. (no exponents) (x x x x)(x x x) Simplify. (x to a power) x7 Can you devise a rule that can be used as a short-cut for finding the product of powers? Slide to check. Multiplying monomials without coefficients Rule: am an = am + n Examples: x4 x7 = x4 + 7 = x11 a3b5 a2b6 = a3+2b5+6 = a5b11 Multiplying monomials with coefficients. 6x3 7x4 regroup the coefficients (6 7) (x3 x4) and the variables. multiply the numbers 42x7 use the rule for multiplying exponents Multiplying Monomials -3x7y3 9x4y6 (-3 9) (x7x4) (y3y6) -27x11y9 Product of Powers W hen multiplying monomials with like bases, add the exponents. Examples 1) 2) 3y7 12a8b8 Slide to check. 23The product of 4ab2 6a3b is 10a4b3. True False 24Multiply A B C D 25What is the coefficient of ? 26 What is the degree of the product of ? 27 A B C D Now, try this. Expanding the expression yields Can you devise a rule that can be used as a short-cut for simplifying a power of a power? Slide to check. Does your rule from the last example apply to this example? Expanding the expression yields Can you refine a rule that can be used as a short-cut for simplifying a power of a product? ( )( )( )= Slide to check. Power of a Power Rule: Examples 2) 1) Slide to check. 28Simplify A B C D 29Simplify A B C D 30Simplify A B C D 31Simplify A B C D Dividing Monomials Return to Table of Contents xxxxxxx xxx Consider this: x7 x3 Write this in expanded notation. Simplify. xxxxxxx xxx x4 = Can you devise a rule that can be used as a short-cut for dividing monomials with like bases? Dividing Monomials with Like Bases Rule: Examples Now, consider this: x3 x5 xxx xxxxx Write this in expanded notation. Simplify = xxx xxxxx 1 x2 If we did this using the short-cut of subtracting the exponents, we'd have: = x3 - 5 = x-2 x3 x5 Since it's impossible to get two different answers for this problem, we conclude that: Negative Exponent Rule For all real numbers,. Examples: Now, consider this: xxxx xxxx Write this in expanded notation. Simplify xxxx xxxx 11 1 = If we did this using the short-cut of subtracting the exponents, we'd have: x4 = x4 - 4 = x0 Since it's impossible to get two different answers for this problem,. Zero Exponent Rule For all non-zero real numbers,. Rules Examples k9 k3 1) h3 h10 3) g8 g 4) d5 2) 6) p5 p18 5) k6 1 g7 1 1 p13 1 h7 Slide to check. Combine like terms. In simplest form, there should be no negative exponents. b5 c2d2 1 g3h11 g-2h-4 gh7 3) m-4n2p5 (mnp8)2 4) m6p11 Slide to check. Examples j-4k9 j-7k-2 1) b0cd-3 b-5c3d-1 2) j3k11 Slide to check. Now, let's try some more problems. 32Simplify A B C D 33Simplify A B C D 34Simplify A B C D 35Simplify A B C D 36Simplify A B C D 37Simplify A B C D 38Simplify A B C D 39Simplify A B C D 40Simplify AB C D x4 y4z5 x0 y4z5 x4 y3z5 1 y4z5 x2y-3z0 x-2yz5 41Simplify AB C D x5y4z0 xy4 x-5y4z0 xy4z0 x-2y4z x-3z 42Simplify A B C D 17 51x4y4z3 1 3y3z3 1 3x4y4z x0y3z3 17x-2y-3 51x2yz3 M ultiplying a Polynomial by a Monomial Return to Table of Contents Find the total area of the rectangles square units To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication. Examples: Simplify. -2x(5x2 - 6x + 8) -2x(5x2 + -6x + 8) (-2x)(5x2) + (-2x)(-6x) + (-2x)(8) -10x3 + 12x x -10x3 + 12x2 - 16x To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication. Examples: Simplify. -3x2(-2x2 + 3x - 12) -3x2(-2x2 + 3x + -12) (-3x2)(-2x2) + (-3x2)(3x) + (-3x2)(-12) 6x4 + -9x3 + 36x2 6x4 - 9x3 + 36x2 12x4y3 - 15x3y4 + 24x2y5 Let's Try It!Multiply to simplify x2(5x2 - 6x - 3) 3.3xy(4x3y2 - 5x2y3 + 8xy4) -2x4 + 4x3 - 7x2 20x4 - 24x3 - 12x Slide to check. 43What is the area of the rectangle shown? A B C D x2 x2 + 2x + 4 44 A B C D 6x2 + 8x x2 + 8x x2 + 8x2 - 12x 6x3 + 8x2 - 12x 45 A B C D 46 A B C D 47Find the area of a triangle (A=1/2bh) with a base of 4x and a height of 2x - 8. All answers are in square units. A B C D Multiplying Polynomials Return to Table of Contents Find the total area of the rectangles sq.units Area of the big rectangle Area of the horizontal rectangles Area of each box sq.units Let us observe the work from the previous example, From to, we changed the problem so that instead of a polynomial times a polynomial, we now have a monomial times a polynomial. Use this to help solve the next example. Find the total area of the rectangles. 2x 4 x 3 To multiply a polynomial by a polynomial, you multiply each term of the first polynomials by each term of the second. Then, add like terms. Example 1: Example 2: (2x + 4y)(3x + 2y) 2x(3x + 2y) + 4y(3x + 2y) 2x(3x) + 2x(2y) + 4y(3x) + 4y(2y) 6x2 + 4xy + 12xy + 8y2 6x2 + 16xy + 8y2 The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of.... First termsOuter terms Inner Terms Last Terms Example: First Outer Inner Last x2 -7x x3 + x2 - 14x -16 Try it!Find each product. 1)(x - 4)(x - 3) 2)(x + 2)(2x2 - 3x - 8) Slide to check. 8x2 - 2xy - 15y2 x4 + x3 - 8x2 + 24x ) (2x - 3y)(4x + 5y) 4) (x2 + 3x - 6)(x2 - 2x + 4) Try it!Find each product. Slide to check. 48What is the total area of the rectangles shown? A B C D 4x 5 2x 4 49 A B C D 50 A B C D 51 A B C D 52 A B C D 53Find the area of a square with a side of A B C D 54What is the area of the rectangle (in square units)? A B C D How would we find the area of the shaded region? Shaded Area = Total area - Unshaded Area sq. units 55What is the area of the shaded region (in sq. units)? A B C D 11x2 + 3x - 8 7x2 + 3x - 9 7x2 - 3x x2 - 3x - 8 56What is the area of the shaded region (in square units)? A B C D 2x2 - 2x - 8 2x2 - 4x - 6 2x2 - 10x - 8 2x2 - 6x - 4 S pecial Binomial Products Return to Table of Contents Square of a Sum (a + b)2 (a + b)(a + b) a2 + 2ab + b2 The square of a + b is the square of a plus twice the product of a and b plus the square of b. Example:(5x + 3)2 (5x + 3)(5x + 3) 25x2 + 30x + 9 Square of a Difference (a - b)2 (a - b)(a - b) a2 - 2ab + b2 The square of a - b is the square of a minus twice the product of a and b plus the square of b. Example:(7x - 4)2 (7x - 4)(7x - 4) 49x2 - 56x + 16 Product of a Sum and a Difference (a + b)(a - b) a2 + -ab + ab + -b2 Notice the -ab and ab a2 - b2equals 0. The product of a + b and a - b is the square of a minus the square of b. Example:(3y - 8)(3y + 8)Remember the inner and 9y2 - 64outer terms equals 0. Try It! Find each product. 1.(3p + 9)29p2 + 54p (6 - p) p + p2 3.(2x - 3)(2x + 3)4x2 - 9 Slide to check. 57 A B C D x x2 + 10x + 25 x2 - 10x + 25 x2 - 25 58 A B C D 59What is the area of a square with sides 2x + 4? A B C D 60 A B C D S olving Equations Return to Table of Contents Given the following equation, what conclusion(s) can be drawn? ab = 0 Since the product is 0, one of the factors, a or b, must be 0. This is known as the Zero Product Property. Zero Product Property Rule: If ab=0, then either a=0 or b=0 Given the following equation, what conclusion(s) can be drawn? (x - 4)(x + 3) = 0 Since the product is 0, one of the factors must be 0. Therefore, either x - 4 = 0 or x + 3 = 0. x - 4 = 0 or x + 3 = x = 4 or x = -3 Therefore, our solution set is {-3, 4}. To verify the results, substitute each solution back into the original equation. (x - 4)(x + 3) = 0 (-3 - 4)(-3 + 3) = 0 (-7)(0) = 0 0 = 0 To check x = -3: (x - 4)(x + 3) = 0 (4 - 4)(4 + 3) = 0 (0)(7) = 0 0 = 0 To check x = 4: What if you were given the following equation? (x - 6)(x + 4) = 0 By the Zero Product Property: x - 6 = 0 or x + 4 = 0 x = 6 x = -4 After solving each equation, we arrive at our solution: {-4, 6} 61Solve (a + 3)(a - 6) = 0. A {3, 6} B {-3, -6} C {-3, 6} D {3, -6} 62Solve (a - 2)(a - 4) = 0. A {2, 4} B {-2, -4} C {-2, 4} D {2, -4} 63Solve (2a - 8)(a + 1) = 0. A {-1, -16} B {-1, 16} C {-1, 4} D {-1, -4} Definitions of Monomials,Polynomials and Degrees Adding and Subtracting Polynomials Multiplying Monomials Dividing by Monomials Multiplying Polynomials Mulitplying a Polynomial by a Monomial Special Binomial Products Solving Equations by Factoring Key Topics