mn 3 17x 2 81y 4 z Polynomials 32,457 Polynomials Part 1 Algebra (MS) Definitions of Monomials, Polynomials and Degrees Table of Contents
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1 Table of ontents Polynomials efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying Monomials ividing by Monomials Mulitplying a Polynomial by a Monomial Multiplying Polynomials Special inomial Products Solving Equations by Factoring efinitions of Monomials, Polynomials and egrees Return to Table of ontents 81y 4 z monomial is a one term expression formed by a number, a variable, or the product of numbers and variables. 17x 2 Examples of monomials... rt 6 4x 32, mn 3
2 rag the following terms into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term w disappear. a + b 5 5x + 7 xy 4 7 x 2 (5 + 7y) 6+5rs 15 Monomials 7x 3 y 5 4 t x 2 yz 3 4(5a 2 bc 2 ) polynomial is an expression that contains two or more monomials. Examples of polynomials a 2 8x 3 +x 2 7+b+c 2 +4d 3 rt 6 c 2 +d a 4 b a 3 2b 2 4c mn 3 egrees 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. 1 What is the degree of? ) 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. constant has a degree 0, because there is no variable.
3 2 What is the degree of? What is the degree of 3? What is the degree of? egrees of Polynomials The degree of a polynomial is the same as that of the term with the greatest degree. 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.
4 Find the degree of each polynomial nswers: 1) 3 1) 0 2) 12c 3 2) 3 3) ab 4) 8s 4 t 5) 2 7n 6) h 4 8t 7) s 3 + 2v 2 y 2 1 3) 2 4) 5 5) 1 6) 4 7) 4 5 What is the degree of the following polynomial: What is the degree of the following polynomial: What is the degree of the following polynomial:
5 8 What is the degree of the following polynomial: dding and Subtracting Polynomials Return to Table of ontents Standard Form The standard form of an equation is to put the terms in order from highest degree to the lowest degree. Standard form is the commonly accepted way to write polynomials. Monomials with the same variables and the same power are like terms. Like Terms Unlike Terms Example: is in standard form. 4x and 12x x3y and 4x3y 3b and 3a 6a2b and 2ab2 Put the following equation into standard form:
6 ombine these like terms using the indicated operation. 9 Simplify 10 Simplify 11 Simplify
7 To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (3x 2 +5x 12) + (5x 2 7x +3) (3x 4 5x) + (7x 4 +5x 2 14x) line up the like terms line up the like terms 3x 2 + 5x 12 3x 4 5x (+)5x 2 7x + 3 (+) 7x 4 +5x 2 14x 8x 2 2x 9 10x 4 +5x 2 19x We can also add polynomials horizontally. (3x x 5) + (5x 2 7x 9) Use the communitive and associative properties to group like terms. (3x 2 + 5x 2 ) + (12x + 7x) + ( 5 + 9) 8x 2 + 5x 14 = 12 dd 13 dd
8 14 dd 15 dd 16 dd To subtract polynomials, subtract the coefficients of like terms. Example: 3x 4x = 7x 13y ( 9y) = 22y 6xy 13xy = 7xy
9 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to addition with multiplication by 1. istribute the 1 and then follow the rules for adding polynomials (3x 2 +4x 5) (5x 2 6x +3) (3x 2 +4x 5) + ( 1)(5x 2 6x+3) (3x 2 +4x 5) + ( 5x 2 +6x 3) 3x 2 + 4x 5 (+) 5x 2 6x + 3 2x 2 +10x 8 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to addition with multiplication by 1. istribute the 1 and then follow the rules for adding polynomials (4x 3 3x 5) (2x 3 +4x 2 7) (4x 3 3x 5) +( 1)(2x 3 +4x 2 7) (4x 3 3x 5) + ( 2x 3 4x 2 +7) 4x 3 3x 5 (+) 2x 3 4x x 3 4x 2 3x + 2 We can also subtract polynomials horizontally. (3x x 5) (5x 2 7x 9) hange the subtraction to adding with multiplication by a negative one and distribute the negative one. (3x x 5) +( 1)(5x 2 7x 9) (3x x 5) + ( 5x 2 + 7x + 9) 17 Subtract Use the commutative and associative properties to group like terms. (3x 2 + 5x 2 ) + (12x +7x) + ( 5 +9) 2x x + 4
10 18 Subtract 19 Subtract 20 Subtract 21 Subtract
11 22 What is the perimeter of the following figure? (answers are in units) Multiplying Monomials Return to Table of ontents Recall: a monomial is a one term expression formed by a number, a variable, or the product of numbers and variable Now that we know how to add and subtract polynomials, we need learn how to multiply them. Let's start with two monomials. Multiplying monomials without coefficients Rule: a m a n = a m + n Write this expression in expanded form. (no exponents) (x Slide x x to x)(x check. x x) Simplify. (x to a power) Slide x 7 to check. Examples: x 4 x 7 = x = x 11 a 3 b 5 a 2 b 6 = a 3+2 b 5+6 = a 5 b 11 an you devise a rule that can be used as a short cut for finding the product of powers?
12 Multiplying Monomials Multiplying monomials with coefficients. 6x 3 7x 4 regroup the coefficients (6 7) (x 3 x 4 ) 3x 7 y 3 9x 4 y 6 ( 3 9) (x 7 x 4 ) (y 3 y 6 ) and the variables. multiply the numbers 42x 7 use the rule for multiplying exponents 27x 11 y 9 Product of Powers When multiplying monomials with like bases, add the exponents. Examples 1) Slide 3yto 7 check. 23 The product of 4ab 2 6a 3 b is 10a 4 b 3. True False 2) 12a 8 b 8 Slide to check.
13 24 Multiply 25 What is the coefficient of? 26 What is the degree of the product of? 27
14 Now, try this. oes your rule from the last example apply to this example? Expanding the expression yields Slide to check. Expanding the expression yields ( )( )( )= Slide to check. an you devise a rule that can be used as a short cut for simplifying a power of a power? an you refine a rule that can be used as a short cut for simplifying a power of a product? Power of a Power Rule: Examples 1) Slide to check. 28 Simplify 2) Slide to check.
15 29 Simplify 30 Simplify 31 Simplify ividing Monomials Return to Table of ontents
16 onsider this: x 7 x 3 Write this in expanded notation. x x x x x x x Simplify. x x x x x x x x x x = x 4 x x x an you devise a rule that can be used as a short cut for dividing monomials with like bases? Examples ividing Monomials with Like ases Rule: Now, consider this: x 3 x 5 x x x Write this in expanded notation. x x x x x Negative Exponent Rule For all real numbers, Simplify x x x 1 = x x x x x x If we did this using the short cut of subtracting the exponents, we'd have: x 3 = x 3 5 = x 2 x 5 Since it's impossible to get two different answers for this problem, we conclude that: Examples:
17 Now, consider this: Write this in expanded notation. x x x x Simplify x x x x x x x x x x x 1 = If we did this using the short cut of subtracting the exponents, we'd have: x 4 x 4 = x 4 4 = x 0 Since it's impossible to get two different answers for this problem, 1 x Zero Exponent Rule For all non zero real numbers, Rules ombine like terms. In simplest form, there should be no negative exponents. Examples 1) k 9 k 3 Slide to check. k 6 4) g 8 g Slide to check. g 7 2) d 5 d 5 Slide 1to check. 5) p 5 p 18 1 Slide to check. p 13 3) h 3 h 10 1 Slide to check. h 7 6) Slide to check. 1
18 Now, let's try some more problems. 32 Simplify Examples 1) j 4 k 9 j 7 k 2 j 3 k 11 Slide to check. 3) g 2 h 4 gh 7 1 Slide to check. g 3 h 11 2) b 0 cd 3 b 5 c 3 d 1 b 5 Slide to check. c 2 d 2 4) (mnp 8 ) 2 Slide m 6 p 11 to check. m 4 n 2 p 5 33 Simplify 34 Simplify
19 35 Simplify 36 Simplify 37 Simplify 38 Simplify
20 39 Simplify 40 Simplify x 4 y 4 z 5 x 0 y 4 z 5 x 2 y 3 z 0 x 2 yz 5 x 4 y 3 z 5 1 y 4 z 5 41 Simplify x 2 y 4 z x 3 z x 5 y 4 z 0 xy 4 x 5 y 4 z 0 xy 4 z 0 42 Simplify x 4 y 4 z x 2 y 3 51x 2 yz 3 3y 3 z x 4 y 4 z 3 51x 0 y 3 z 3
21 Find the total area of the rectangles. 3 Multiplying a Polynomial by a Monomial square units Return to Table of ontents square units To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponen multiplication. Examples: Simplify. 2x(5x 2 6x + 8) 2x(5x 2 + 6x + 8) ( 2x)(5x 2 ) + ( 2x)( 6x) + ( 2x)(8) 10x x x 10x x 2 16x To multiply a polynomial by a monomial, you use the distributive property together with the laws of expone multiplication. Examples: Simplify. 3x 2 ( 2x 2 + 3x 12) 3x 2 ( 2x 2 + 3x + 12) ( 3x 2 )( 2x 2 ) + ( 3x)(3x) + ( 3x)( 12) 6x 4 + 9x x 6x 4 9x x
22 Let's Try It! Multiply to simplify. 43 What is the area of the rectangle shown? 1. 2x Slide 4 + to 4x check. 3 7x 2 x 2 x2 + 2x x 2 (5x 2 6x 3) 20x 4 24x 3 12x Slide to check. 3. 3xy(4x 3 y 2 5x 2 y 3 + 8xy 4 ) 12x 4 y 3 15x 3 y x 2 y 5 Slide to check x 2 + 8x 12 6x 2 + 8x x 2 + 8x 2 12x 6x 3 + 8x 2 12x
23 46 47 Find the area of a triangle (= 1 / 2bh) with a base of 4x and a height of 2x 8. ll answers are in square units. Find the total area of the rectangles Multiplying Polynomials 6 Return to Table of ontents sq.units
24 Let us observe the work from the previous example, Find the total area of the rectangles. x 2x 4 sq.units 3 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. 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: (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 terms Outer terms Inner Terms Last Terms Example: First Outer Inner Last Example 2:
25 Try it! 1) (x 4)(x 3) Find each product. x 2 Slide 7x + to 12check. Try it! Find each product. 3) (2x 3y)(4x + 5y) 8x Slide 2 2xy to 15y check. 2 2) (x + 2)(2x 2 3x 8) 2xSlide 3 + x 2 to 14x check. 16 4) (x 2 + 3x 6)(x 2 2x + 4) Slide to check. x 4 + x 3 8x x What is the total area of the rectangles shown? 49 2x 4 4x 5
26 Find the area of a square with a side of
27 54 What is the area of the rectangle (in square units)? How would we find the area of the shaded region? Shaded rea = Total area Unshaded rea 3x 2 + 5x + 2 3x 2 + 6x + 2 4x 2 + 5x + 2 3x sq. units 55 What is the area of the shaded region (in sq. units)? 56 What is the area of the shaded region (in square units)? 11x 2 + 3x 8 2x 2 2x 8 7x 2 + 3x 9 2x 2 4x 6 7x 2 3x 10 2x 2 10x 8 11x 2 3x 8 2x 2 6x 4
28 Square of a Sum Special inomial Products (a + b) 2 (a + b)(a + b) a 2 + 2ab + b 2 The square of a + b is the square of a and b plus the square of b. a plus twice the product of Return to Table of ontents Example: (5x + 3) 2 (5x + 3)(5x + 3) 25x x + 9 Square of a ifference (a b) 2 (a b)(a b) a 2 2ab + b 2 The square of a b is the square of a and b plus the square of b. Example: (7x 4) 2 (7x 4)(7x 4) 49x 2 56x + 16 a minus twice the product of Product of a Sum and a ifference (a + b)(a b) a + ab + ab + b Notice the ab and ab 2 2 a b equals 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 9y 2 64 outer terms equals 0.
29 Try It! Find each product (3p + 9) 2 9p 2 + Slide 54p to + check (6 p) 2 36 Slide 12p to + check. p 2 3. (2x 3)(2x + 3) 4xSlide 2 9to check. x x x + 25 x 2 10x + 25 x What is the area of a square with sides 2x + 4?
30 60 Solving Equations Return to Table of ontents 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. Zero Product Property Rule: If ab=0, then either a=0 or b=0 This is known as the Zero Product Property.
31 Given the following equation, what conclusion(s) can be dra (x 4)(x + 3) = 0 Since the product is 0, one of the factors must be 0. Therefore, eitherx 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 solution back into the original equation. To check x = 3: (x 4)(x + 3) = 0 To check x = 4: (x 4)(x + 3) = 0 ( 3 4)( 3 + 3) = 0 ( 7)(0) = 0 0 = 0 (4 4)(4 + 3) = 0 (0)(7) = 0 0 = 0 What if you were given the following equation? y the Zero Product Property: (x 6)(x + 4) = 0 x 6 = 0 or x + 4 = 0 x = 6 x = 4 fter solving each equation, we arrive at our solution: { 4, 6} 61 Solve (a + 3)(a 6) = Solve (a 2)(a 4) = 0. {2, 4} {3, 6} { 3, 6} { 3, 6} {3, 6} { 2, 4} { 2, 4} {2, 4}
32 63 Solve (2a 8)(a + 1) = 0. { 1, 16} { 1, 16} { 1, 4} { 1, 4} Key Topics efinitions of Monomials,Polynomials and egrees dding and Subtracting Polynomials Multiplying Monomials ividing by Monomials Mulitplying a Polynomial by a Monomial Multiplying Polynomials Special inomial Products Solving Equations by Factoring
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