Monomials. Polynomials. Objectives: Students will multiply and divide monomials Students will solve expressions in scientific notation

Transcription

1 Students will multiply and divide monomials Students will solve expressions in scientific notation 5.1 Monomials Many times when we analyze data we work with numbers that are very large. To simplify these large numbers, these numbers may be written using scientific notation or using exponents. A monomial is an expression that is a number, a variable, or the product of a number and one or more variables. Monomials cannot contain variables in denominators, variables with exponents that are negative, or variables under radical signs. Monomials Non- Monomials 5b, -w, 23, x 2, 1 3 x3 y 4 1 n 4, 3 x, x + 8, a 1

2 Students will multiply and divide monomials Students will solve expressions in scientific notation 5.1 Monomials Constants are monomials that contain no variables. Coefficients are numerical factors of variables. The degree of a monomial is the sum of all the exponents of the variables. The degree of 12g 7 h 4 is 11. The degree of a constant is 0. A power is an expression of the form x n. The word power is also used to refer to the exponent itself. Negative exponents express the multiplicative inverse of a number. x 2 = 1 x 2 Is x 2 a monomial? No, why?

3 Students will multiply and divide monomials Students will solve expressions in scientific notation 5.1 Monomials For any real number a 0 and any integer n, a n = 1 1 an and = a n an. 2 3 = and 1 z 5 = z5 For any real number a and integers m and n, a m a n = a m+n. x 2 x 3 = x x x x x = x 2+3 = x 5 For any real number a 0, and integers m and n, am a n = am n. g 7 g g g g g g g g4 = g g g g Show that x 0 = 1; x 5 = g 7 4 = g 3 ; h3 h 8 = h h h h h h h h h h h = 1 h 5 = h 5 x 5 = x5 5 = x 0 and x5 x x x x x x5 = x x x x x = 1 therefore x 0 = 1

4 5.1 Monomials Students will multiply and divide monomials Students will solve expressions in scientific notation Power of a Power: a m n = a mn a 3 2 = a 3 a 3 = a 3+3 = a 6 Power of a Product: ab m = a m b m ab 3 = ab ab ab = a a a b b b = a 3 b 3 Power of a Quotient: a b n = a n b n a b 4 a = 4 ; a b 4 b 3 = b a 3 = b 3 a 3

5 5.1 Monomials Students will multiply and divide monomials Students will solve expressions in scientific notation Very large and very small numbers written in standard notation can be written as scientific notation; in the form a 10 n, where 1 a < 10 and n is an integer. 6,380,000 = ; = = = = = Dimensional Analysis uses units with the numbers. If units are given, the answer must have units. After the sun, Alpha Centauri C is the closest star to Earth which is meters away. How long does it take light from Alpha Centauri C to reach Earth? d = rt; where d is distance, r is rate, and t is time t = d r = meters meters/sec sec or 4.2 years Bookwork: page 226; problems even

6 Students will add and subtract polynomials Students will multiply polynomials 5.2 Polynomials Shenequa wants to attend an out-of-state university where the tuition is $8820. The tuition increases at a rate of 4% per year. Polynomials can be used to represent this increase in tuition. If r represents the rate of increase, then the tuition for the second year will be 8820(1+r). The third year tuition is 8820(1 + r) 2, or 8820r ,640r when expanded. A Polynomial is a monomial or a sum of monomials. The monomials that make up a polynomial are called the terms of the polynomial. Remember we can collect like terms in polynomials. A polynomial with three terms is called a trinomial, e.g. x 2 + 3x + 1; while xy + z 3 is a binomial. The degree of the polynomial is the degree of the monomial with the greatest degree.

7 Students will add and subtract polynomials Students will multiply polynomials 5.2 Polynomials To simplify a polynomial means to perform the operations indicated and combine like terms. 3x 2 2x + 3 (x 2 + 4x 2) 3x 2 2x + 3 x 2 4x + 2 2x 2 6x + 5 2x(7x 2 3x + 5) 14x 3 6x x We use the distributive property when multiplying polynomials. 3y + 2 5y + 4 = 3y 5y + 3y y y y + 8 This is called the FOIL method; Firsts, Outers, Inners, and Lasts.

8 Students will add and subtract polynomials Students will multiply polynomials 5.2 Polynomials The Vertical Method can also be used. 34 x x y + 2 5y y y 12y y y + 8 Bookwork: page 231; problems even, and even

9 5.3 Dividing Polynomials Students will divide polynomials using long division Students will divide polynomials using synthetic division Remember long division: 3248 divided by Answer: ( 12) Remainder of 8 24 = 1 3

10 Students will divide polynomials using long division Students will divide polynomials using synthetic division 5.3 Dividing Polynomials In lesson 5.1 we learned how to divide monomials. We can also divide a polynomial by a monomial. Simplify 4x 3 y 2 +8xy 2 12x 2 y 3 4xy = 4x3 y 2 4xy + 8xy2 12x2 y 3 4xy 4xy = x 2 y + 2y 3xy 2 The division algorithm can be used to divide a polynomial by a polynomial. Use long division to find z 2 + 2z 24 z 4 z + 6 z 4 z 2 + 2z 24 z 2 4z 6z 24 6z 24 The remainder is zero.

11 5.3 Dividing Polynomials Students will divide polynomials using long division Students will divide polynomials using synthetic division Simplify: t 2 + 3t 9 5 t 1 This is a division problem, isn t it? t 8 t + 5 t 2 + 3t 9 t 2 5t 8t 9 8t Answer: t t

12 Students will divide polynomials using long division Students will divide polynomials using synthetic division 5.3 Dividing Polynomials Lets see what synthetic division looks like: 5x 3 13x x 8 x 2 Step 1: Step 2: Write the coefficients of the dividend in descending order of the degree. Write the constant to the left and bring down the first coefficient. The divisor must be in the form of x r Step 3: Step 4: Multiply the first coefficient by the constant, then add to the second coefficient. Multiply the sum by the constant, then add to the next coefficient. Step 5: Continue until the remainder is determined. The numbers on the bottom row are the coefficients of the quotient. Start with the power of x that is one degree less. 5x 2 3x + 4

13 5.3 Dividing Polynomials Students will divide polynomials using long division Students will divide polynomials using synthetic division Synthetic division works only when the divisor is in the form x r. If the coefficient on the variable is not 1, the divisor must be rewritten. (8x 4 4x 2 + x + 4) (2x + 1) Divide the numerator and denominator by the divisor s coefficient. (4x 4 2x x + 2) (x ) Bookwork: page 236; problems even 3 4x 3 2x 2 x x x = 3 2 2x = x + 1 4x 3 2x 2 x x + 1

14 Students will factor polynomials Students simplify polynomial quotients by factoring 5.4 Factoring Polynomials We have seen where factoring an expression can simplify the expression 8a 3 b 2 + 4a 2 b 3 2ab 2ab = 2ab(4a2 b + 2ab 2 1) 2ab Polynomials can be factored the same way using Factoring Techniques. Number of Terms Factoring Technique Examples = 4a 2 b + 2ab 2 1 Any number Greatest Common Factor (GCF) a 3 b 2 + 2a 2 b 4ab 2 = ab(a 2 b + 2a 4b) Two Three Difference of two Squares Sum of two Cubes Difference of two Cubes Perfect Square Trinomials General Trinomials a 2 b 2 = (a + b)(a b) a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b 2 ) a 2 + 2ab + b 2 = a + b 2 a 2 2ab + b 2 = a b 2 acx 2 + ad + bc x + bd = (ax + b)(cx + d) Four or more Grouping ax + bx + ay + by = x a + b + y a + b = (a + b)(x + y)

15 Students will factor polynomials Students simplify polynomial quotients by factoring Factor each expression. 5.4 Factoring Polynomials 3xy 2 48x = 3x(y 2 16) = 3x(y + 4)(y 4) c 3 d = cd = cd + 3 c 2 d 2 3cd + 9 m 6 n 6 This could be a difference of squares or a difference of cubes. Difference of squares should be done first to make the next step easier. = m 3 + n 3 m 3 n 3 = m + n m 2 mn + n 2 m n m 2 + mn + n 2 5x 2 13x + 6 Use the reverse FOIL method. The coefficients must be two numbers whose product is 5 6 and whose sum is and 10 5x 3 x 2 Bookwork: page 242; problems even

16 Students will simplify radicals Students will use a calculator to approximate radicals 5.5 Roots or Real Numbers Does everyone know why we call it squaring a number or cubing a number, while the other powers do not have a name? When we square a number, we find the area of a square. a = a 2 a When we cube a number, we find the volume of a cube. b b b = b 3

17 Students will simplify radicals Students will use a calculator to approximate radicals 5.5 Roots or Real Numbers What are we doing when we square a number? What about when we cube a number? Multiply that number by itself. Multiply that number by itself three times. If division is the opposite of multiplication, can we do the opposite of squaring or cubing? Yes, we call this, finding the roots of a number, e.g. square root, cubed root, etc. Can we perform any algebraic operation to find the roots of numbers? No, either we know the roots or we do not know. A calculator helps. Exponents help greatly. What is the fourth root of ab 4? ab For any real numbers a and b, if a 2 = b, then a is a square root of b. Since 5 2 = 25, 5 is a square root of 25. For any real numbers a and b, and any positive integer n, if a n = b, then a is an nth root of b. Since 2 5 = 32, 2 is a fifth root of 32.

18 Students will simplify radicals Students will use a calculator to approximate radicals 5.5 Roots or Real Numbers A new symbol, called the radical symbol, is used to indicate the nth root of a number. Some numbers have more than one real nth root. For example, 36 has two square roots, 6 and -6. When there is more than one real root, the nonnegative root is known as the principle root. Only when indicated by an index, are we interested in the negative root. n or ± If n is odd and b is negative, n b, there will be no positive root. The principle root is negative.

19 n Polynomials Students will simplify radicals Students will use a calculator to approximate radicals n b if b > Roots or Real Numbers n b if b < 0 b = 0 even one positive, one negative root no real roots One real root, 0 odd one positive root, no negative roots no positive roots, one negative root ± 25x 4 = ± 5x 2 2 y = y = ±5x 2 = y x 15 y 20 = 5 2x 3 y 4 5 = 2x 3 y 4 Therefore, this does not have a real 9 n is even and b is negative root. 0

20 Students will simplify radicals Students will use a calculator to approximate radicals 5.5 Roots or Real Numbers Remember, when no index is given, we are finding the principle root. If n is even, then the principle root is a positive number. If the nth root of an even power results in an odd power, you must take the absolute value of the result. 5 2 = 5 = 5 = = 2 3 If the result is an even power or you find the nth root of an odd power, there is no need to take the absolute value. Why? = 8 Bookwork: page 248; problems even; look at 58-62

21 Students will simplify radical expressions Students will add, subtract, multiply, and divide radical expressions 5.6 Radical Expressions How do we simplify the expression 3 5? = For any real numbers a and b, and any integer n > 1, then = 3 5 therefore; 3 5 = 15 If n is even and a and b are both nonnegative, n ab = n a n b If n is odd, n ab = n a n b Though these two rules look the same, the second rule allows for negative vales of a and b. 16p 8 q 7 = 42 p 4 2 q 3 2 q = 4p 4 q 3 q However, for this to be defined 16p 8 q 7 must be nonnegative; meaning, q must be nonnegative.

22 5.6 Radical Expressions Students will simplify radical expressions Students will add, subtract, multiply, and divide radical expressions What about division? Consider, 49. This is a perfect square, so 49 = = 7 3 For any real numbers a and b 0, and any integer n > 1, then n a b = n a n b How do we know a radical expression is in simplest form? The index n is as small as possible. The radicand contains no factors (other than 1) that are nth powers of an integer or polynomial. The radicand contains no fractions. No radicals appear in the denominator. Wait! What do you mean, no radicals in the denominator? How do you get rid of radicals in the denominator?

23 Students will simplify radical expressions Students will add, subtract, multiply, and divide radical expressions 5.6 Radical Expressions To eliminate radicals from the denominator, we rationalize the denominator. Remember when we have a fraction in the denominator we multiply by a form of 1 to eliminate the fraction. We do the same thing with radicals. x 4 = x4 = y 5 y 5 = = x2 y 2 x 2 2 y 2 2 y x 2 2 y 2 2 y = x2 y y 2 y y y 5 5 4a = a 5 40a 4 = 5 32a a 4 = 2a 5 8a 4 5 8a 4 WHY?

24 Students will simplify radical expressions Students will add, subtract, multiply, and divide radical expressions We now know that 2 2 = 2, does = 2? 5.6 Radical Expressions Given: 1 c What is c =? From geometry, a 2 + b 2 = c 2 c = If we double the triangle, can = 2? c No, because the hypotenuse MUST be longer than any one side. 1 2

25 Students will simplify radical expressions Students will add, subtract, multiply, and divide radical expressions 5.6 Radical Expressions Adding radicals is like adding monomials. We must combine like terms. Like radical expressions are alike if The indices are alike. The radicands are alike. 3 and 3 3 are not alike 4 5 and 4 5x are not alike 2 4 3a and 5 4 3a are like expressions = = = 3 3

26 Students will simplify radical expressions Students will add, subtract, multiply, and divide radical expressions (5 3 6)( ) = FOIL = = Radical Expressions Notice that a b + c d and a b c d are conjugates. The product of conjugates is always a rational number. Conjugates are used to rationalize denominators. Bookwork: page254; problems even = = =

27 Students will write radical expressions using exponents. Students will simplify expressions in radical or exponent form. 5.7 Rational Exponents In lesson 5.5, we determined that squaring a number and taking the square root of a number are inverse operations. Does this mean we can express a radical as an exponent? b = b 1 2 b 1 2 Yes! = b = b This means that b 1 2 is a number whose square is b. Therefore, b 1 2 = b For any real number b and for any positive integer n, b 1 n = n b, except when b < 0 and n is even.

28 Students will write radical expressions using exponents. Students will simplify expressions in radical or exponent form. 5.7 Rational Exponents Evaluate each expression = = = = = = 2 1 = 1 2 = 1 2 Is there another way to evaluate this expression. Remember, exponent rules allow us to do many different things. Allowing us to use our strengths.

29 Students will write radical expressions using exponents. Students will simplify expressions in radical or exponent form. Evaluate each expression. 5.7 Rational Exponents = = = = 3 3 = 27 = = = = 27

30 Students will write radical expressions using exponents. Students will simplify expressions in radical or exponent form. The last example leads to the following rule. 5.7 Rational Exponents For any nonzero real number b, and any integers m and n, with n > 1, b m n = n b m = n b m, except when b < 1 and n is even. How do we know a rational expression is simplified? No negative exponents. No fractional exponents in the denominator. Not a complex fraction. The index of any remaining radical is the least number possible. Bookwork: page 261; problems even

31 Students will solve equations containing radicals. Students will solve inequalities containing radicals. 5.8 Radical Equations and Inequalities A computer chip manufacturer has determined that the cost to manufacture their chips is C = 10n This formula has a radical in it. Equations that have variables in the radicand are called radical equations. To solve this type of equation, isolate the radicand and then raise each side of the equation to the power of the index of the radical to eliminate the radical. x = 4 x + 1 = 2 x = 2 2 x + 1 = 4 x = 3 We should always check our solution. Sometimes we will obtain a solution that does not satisfy the equation. This solution is called an extraneous solution. x 15 = 3 x x 15 2 = 3 x 2 x 15 = 9 6 x + x 24 = 6 x 4 = x 4 2 = x 2 16 = x

32 Students will solve equations containing radicals. Students will solve inequalities containing radicals. 5.8 Radical Equations and Inequalities Lets check this solution. x 15 = 3 x = If we graph y = x 15 and y = 3 x on our calculators, we see the two graphs do not intersect; meaning, there is no solution. 3 5n = 0 3 5n = 2 5n = 2 3 5n = n 1 = n = n = 7 27 If we check this solution, we find 7 is a solution. 27

33 Students will solve equations containing radicals. Students will solve inequalities containing radicals. 5.8 Radical Equations and Inequalities Knowing this information, we can solve radical inequalities. A radical inequality is an inequality with a radicand x 4 6 First, the radicand must be greater than or equal to zero. Now we must solve the original inequality. 4x 4 0 x x 4 6 4x 4 4 4x 4 16 x 5 It appears our solutions are 1 x 5. By solving for f(0), f(2), and f(7) we can verify this.

34 Students will solve equations containing radicals. Students will solve inequalities containing radicals. 5.8 Radical Equations and Inequalities To solve radical inequalities, use the following steps Step 1: if the index of the root is even, identify the values of the variable for which the radicand is nonnegative. Step 2: Step 3: solve the inequality algebraically. test values to check the solution or solution set. Bookwork: page 266; problems even

35 Students will add and subtract complex numbers. Students will multiply and divide complex numbers. 5.9 Complex Numbers When we solve the equation 2x = 0, we find that x 2 = 1. This is not a real solution; however, many solutions of radicands have a negative solution. Rene Descartes proposed that the number i be defined such that i 2 = 1. This means that i = 1. This is called the imaginary unit. Numbers in the form of 3i, -5i, and i 2 are pure imaginary numbers. Pure imaginary numbers are square roots of negative real numbers. b 2 = b 2 1 = bi 18 = = 3i 2 125x 5 = x 4 5x = 5ix 2 5x

36 Students will add and subtract complex numbers. Students will multiply and divide complex numbers. 5.9 Complex Numbers 2i 7i = 14i 2 = 14 1 = 14 i 45 = i i 44 = i i 2 22 = i 1 22 = i = i 10 i 15 = i = = 5 6 3x = 0 3x 2 = 48 x 2 = 16 x = ± 16 x = ±4i

37 Students will add and subtract complex numbers. Students will multiply and divide complex numbers. 5.9 Complex Numbers What about the expression 5 + 2i. Since 5 is a real number and 2i is an imaginary number, the terms are not like terms. This expression is called a complex number. A complex number is any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. a is the real part and b is called the imaginary part. Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal. 2x 3 + y 4 i = 3 + 2i 2x 3 = 3 and y 4 = 2 x = 3 and y = 6 Can we solve this when we have two variables and one equation? Yes, because x is a real part and y is an imaginary part; giving us two variables and two equations.

38 Students will add and subtract complex numbers. Students will multiply and divide complex numbers. 5.9 Complex Numbers In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I Z. Find the voltage in a circuit with current 1 + 3j amps and the impedance 7 5j ohms. E = I Z E = (1 + 3j) (7 5j) E = 7 5j + 21j 15j 2 = j + 15 = j volts Two complex number in the form of a + bi and a bi are complex conjugates. The product of complex conjugate is always a real number. This fact can be used to simplify the quotient of two complex numbers. 3i 2 + 4i = 3i 2 + 4i 2 4i 2 4i = 6i 12i2 4 16i 2 = 6i = i Bookwork: page 274; problems even Remember, a + bi is standard form for imaginary numbers.