Polynomials 5.1 Monomials Objectives: • Students will multiply and divide monomials • Students will solve expressions in scientific notation 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, 2 , 1 3 3 4 1 4 , 3 , + 8, −1
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Polynomials5.1 Monomials
Objectives:• Students will multiply and divide monomials• Students will solve expressions in scientific notation
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, 𝑥2,1
3𝑥3𝑦4 1
𝑛4 , 3 𝑥, 𝑥 + 8, 𝑎−1
Polynomials5.1 Monomials
Objectives:• Students will multiply and divide monomials• Students will solve expressions in scientific notation
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 12𝑔7ℎ4 𝑖𝑠 11. The degree of a constant is 0.
A power is an expression of the form 𝑥𝑛. The word power is also used to refer to the exponent itself.
Negative exponents express the multiplicative inverse of a number. 𝑥−2 =1
𝑥2
Is 𝑥−2 a monomial? No, why?
Polynomials5.1 Monomials
Objectives:• Students will multiply and divide monomials• Students will solve expressions in scientific notation
For any real number 𝑎 ≠ 0 and any integer n, 𝑎−𝑛 =1
𝑎𝑛 𝑎𝑛𝑑1
𝑎−𝑛 = 𝑎𝑛.
2−3 =1
23 𝑎𝑛𝑑1
𝑧−5 = 𝑧5
For any real number a and integers m and n, 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛.
𝑥2 ∙ 𝑥3 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 = 𝑥2+3 = 𝑥5
For any real number 𝑎 ≠ 0, and integers m and n, 𝑎𝑚
𝑎𝑛 = 𝑎𝑚−𝑛.
𝑔7
𝑔4 =𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔
𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔= 𝑔7−4 = 𝑔3;
ℎ3
ℎ8 =ℎ ∙ ℎ ∙ ℎ
ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ=
1
ℎ5 = ℎ−5
Show that 𝑥0 = 1;
𝑥5
𝑥5 = 𝑥5−5 = 𝑥0 and𝑥5
𝑥5 =𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥
𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥= 1 therefore 𝑥0 = 1
Polynomials5.1 Monomials
Objectives:• Students will multiply and divide monomials• Students will solve expressions in scientific notation
Objectives:• 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 × 10𝑛, where 1 ≤ 𝑎 < 10 and n is an integer.
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 4 × 1016 meters away. How long does it take light from Alpha Centauri C to reach Earth?
Objectives:• Students will add and subtract polynomials• Students will multiply 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 8820𝑟2 + 17,640𝑟 + 8820 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. 𝑥2 + 3𝑥 + 1; while 𝑥𝑦 + 𝑧3 is a binomial.
The degree of the polynomial is the degree of the monomial with the greatest degree.
Polynomials5.2 Polynomials
Objectives:• Students will add and subtract polynomials• Students will multiply polynomials
To simplify a polynomial means to perform the operations indicated and combine like terms.
3𝑥2 − 2𝑥 + 3 − (𝑥2 + 4𝑥 − 2)
3𝑥2 − 2𝑥 + 3 − 𝑥2 − 4𝑥 + 2
2𝑥2 − 6𝑥 + 5
2𝑥(7𝑥2 − 3𝑥 + 5)
14𝑥3 − 6𝑥2 + 10𝑥
We use the distributive property when multiplying polynomials.
3𝑦 + 2 5𝑦 + 4 = 3𝑦 ∙ 5𝑦 + 3𝑦 ∙ 4 + 2 ∙ 5𝑦 + 2 ∙ 4
15𝑦2 + 22𝑦 + 8
This is called the FOIL method; Firsts, Outers, Inners, and Lasts.
Polynomials5.2 Polynomials
Objectives:• Students will add and subtract polynomials• Students will multiply polynomials
The Vertical Method can also be used.
3𝑦 + 2
× 5𝑦 + 4
12𝑦 + 8
15𝑦2 + 10𝑦
15𝑦2 + 22𝑦 + 8
Bookwork: page 231; problems 16-32 even, and 38-50 even
34x 52
30 + 4x 50 + 2
60 + 8
1500 + 200 + 00 + 0
1768
Polynomials5.3 Dividing Polynomials
Objectives:• Students will divide polynomials using long division• Students will divide polynomials using synthetic division
Remember long division: 3248 divided by 24.
3000 + 200 + 40 + 820 + 4
100
− 2000 + 400
1000 – 200 + 40
+ 50
− 1000 + 200
−400 + 40 + 8
− 20
− − 400 − 80
120 + 8
+ 5
− 100 + 20
20 + (−12) Remainder of 8
24=
1
3
Answer: 135 1
3
Polynomials5.3 Dividing Polynomials
Objectives:• Students will divide polynomials using long division• Students will divide polynomials using synthetic division
In lesson 5.1 we learned how to divide monomials. We can also divide a polynomial by a monomial.
Simplify 4𝑥3𝑦2+8𝑥𝑦2−12𝑥2𝑦3
4𝑥𝑦=
4𝑥3𝑦2
4𝑥𝑦+
8𝑥𝑦2
4𝑥𝑦−
12𝑥2𝑦3
4𝑥𝑦
= 𝑥2𝑦 + 2𝑦 − 3𝑥𝑦2
The division algorithm can be used to divide a polynomial by a polynomial.
Use long division to find 𝑧2 + 2𝑧 − 24 ÷ 𝑧 − 4
𝑧 − 4 𝑧2 + 2𝑧 − 24
𝑧
− 𝑧2 − 4𝑧
6𝑧 − 24
+ 6
− 6𝑧 − 24The remainder is zero.
Polynomials5.3 Dividing Polynomials
Objectives:• Students will divide polynomials using long division• Students will divide polynomials using synthetic division
Simplify: 𝑡2 + 3𝑡 − 9 5 − 𝑡 −1
𝑡2 + 3𝑡 − 9−𝑡 + 5
−𝑡
− 𝑡2 − 5𝑡
8𝑡 − 9
− 8
− 8𝑡 − 40
31
This is a division problem, isn’t it?
Answer: −𝑡 − 8 +31
5−𝑡
5.3 Dividing PolynomialsPolynomials
Objectives:• Students will divide polynomials using long division• Students will divide polynomials using synthetic division
Lets see what synthetic division looks like:5𝑥3 − 13𝑥2 + 10𝑥 − 8 ÷ 𝑥 − 2
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
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.
5 − 13 10 − 8
5
Multiply the first coefficient by the constant, then add to the second coefficient.
2
10
-3
Multiply the sum by the constant, then add to the next coefficient.
-6
4
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.
8
0
5𝑥2 − 3𝑥 + 4
5.3 Dividing PolynomialsPolynomials
Objectives:• 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.
(8𝑥4 − 4𝑥2 + 𝑥 + 4) ÷ (2𝑥 + 1)
Divide the numerator and denominator by the divisor’s coefficient.
(4𝑥4 − 2𝑥2 +1
2𝑥 + 2) ÷ (𝑥 +
1
2)
−1
24 0 -2
1
22
-2 1 1
2−
1
2
4 0 -1 1 3
2
4𝑥3 − 2𝑥2 − 𝑥 + 1 +
32
𝑥 +12
3
2÷ 𝑥 +
1
2=
3
2÷
2𝑥 + 1
2=
3
2∙
2
2𝑥 + 1
4𝑥3 − 2𝑥2 − 𝑥 + 1 +3
2𝑥 + 1Bookwork: page 236; problems 16-50 even
Polynomials5.4 Factoring Polynomials
Objectives:• Students will factor polynomials• Students simplify polynomial quotients by factoring
We have seen where factoring an expression can simplify the expression…
8𝑎3𝑏2 + 4𝑎2𝑏3 − 2𝑎𝑏
2𝑎𝑏=
2𝑎𝑏(4𝑎2𝑏 + 2𝑎𝑏2 − 1)
2𝑎𝑏= 4𝑎2𝑏 + 2𝑎𝑏2 − 1
Polynomials can be factored the same way using Factoring Techniques.
Number of Terms Factoring Technique Examples
Any number Greatest Common Factor (GCF) 𝑎3𝑏2 + 2𝑎2𝑏 − 4𝑎𝑏2 = 𝑎𝑏(𝑎2𝑏 + 2𝑎 − 4𝑏)
Two Difference of two SquaresSum of two CubesDifference of two Cubes
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.
1 − 3
5 + 3∙
5 − 3
5 − 3=
5 − 3 − 5 3 + 3
25 − 3
=8 − 6 3
22
=4 − 3 3
11Bookwork: page254; problems 16-48 even
Polynomials5.7 Rational Exponents
Objectives:• Students will write radical expressions using exponents.• Students will simplify expressions in radical or exponent form.
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? Yes!
𝑏12
2
= 𝑏12 ∙ 𝑏
12
= 𝑏12+
12
= 𝑏
This means that 𝑏1
2 is a number whose square is 𝑏. Therefore, 𝑏1
2 = 𝑏
For any real number b and for any positive integer n, 𝑏1
𝑛 =𝑛
𝑏, except when 𝑏 < 0 and n is even.
Polynomials5.7 Rational Exponents
Objectives:• Students will write radical expressions using exponents.• Students will simplify expressions in radical or exponent form.
Evaluate each expression.
16−14 =
1
1614
=1
416
=1
424
=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.
= 24 −14
= 24 −14
= 2−1
=1
2
Polynomials5.7 Rational Exponents
Objectives:• Students will write radical expressions using exponents.• Students will simplify expressions in radical or exponent form.
Evaluate each expression.
24335 = 2433
15
=5
2433
=5
35 3
=5
35 ∙ 35 ∙ 35
= 3535
= 33
= 27
= 3 ∙ 3 ∙ 3
= 27
Polynomials5.7 Rational Exponents
Objectives:• Students will write radical expressions using exponents.• Students will simplify expressions in radical or exponent form.
The last example leads to the following rule.
For any nonzero real number b, and any integers m and n, with 𝑛 > 1, 𝑏𝑚
𝑛 =𝑛
𝑏𝑚 =𝑛
𝑏𝑚
, 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 22-64 even
Polynomials5.8 Radical Equations and Inequalities
Objectives:• Students will solve equations containing radicals.• Students will solve inequalities containing radicals.
A computer chip manufacturer has determined that the cost to manufacture their chips is
𝐶 = 10𝑛2
3 + 1500. 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 + 1 + 2 = 4 𝑥 + 1 = 2 𝑥 + 12
= 22 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 − 152
= 3 − 𝑥 2 𝑥 − 15 = 9 − 6 𝑥 + 𝑥
−24 = −6 𝑥 4 = 𝑥 42 = 𝑥 2 16 = 𝑥
Polynomials5.8 Radical Equations and Inequalities
Objectives:• Students will solve equations containing radicals.• Students will solve inequalities containing radicals.
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 − 113 − 2 = 0 3 5𝑛 − 1
13 = 2 5𝑛 − 1
13 =
2
3
5𝑛 − 113
3
=2
3
35𝑛 − 1 =
8
275𝑛 =
35
27
𝑛 =7
27If we check this solution, we find
7
27is a solution.
Polynomials5.8 Radical Equations and Inequalities
Objectives:• Students will solve equations containing radicals.• Students will solve inequalities containing radicals.
Knowing this information, we can solve radical inequalities. A radical inequality is an inequality with a radicand.
2 + 4𝑥 − 4 ≤ 6 First, the radicand must be greater than or equal to zero.
4𝑥 − 4 ≥ 0 𝑥 ≥ 1
Now we must solve the original inequality.
2 + 4𝑥 − 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.
Polynomials5.8 Radical Equations and Inequalities
Objectives:• Students will solve equations containing radicals.• Students will solve inequalities containing radicals.
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: solve the inequality algebraically.
Step 3: test values to check the solution or solution set.
Bookwork: page 266; problems 14-38 even
Polynomials5.9 Complex Numbers
Objectives:• Students will add and subtract complex numbers.• Students will multiply and divide complex numbers.
When we solve the equation 2𝑥2 + 2 = 0, we find that 𝑥2 = −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 𝑖2 = −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.
−𝑏2 = 𝑏2 ∙ −1 = 𝑏𝑖
−18 = −1 ∙ 32 ∙ 2
= 3𝑖 2
−125𝑥5 = −1 ∙ 52 ∙ 𝑥4 ∙ 5𝑥
= 5𝑖𝑥2 5𝑥
Polynomials5.9 Complex Numbers
Objectives:• Students will add and subtract complex numbers.• Students will multiply and divide complex numbers.
−2𝑖 ∙ 7𝑖 = −14𝑖2
= −14 ∙ −1
= 14
−10 ∙ −15 = 𝑖 10 ∙ 𝑖 15
= 𝑖2 150
= −1 ∙ 25 ∙ 6
= −5 6
𝑖45 = 𝑖 ∙ 𝑖44
= 𝑖 ∙ 𝑖2 22
= 𝑖 ∙ −1 22
= 𝑖
3𝑥2 + 48 = 0
3𝑥2 = −48
𝑥2 = −16
𝑥 = ± −16
𝑥 = ±4𝑖
Polynomials5.9 Complex Numbers
Objectives:• Students will add and subtract complex numbers.• Students will multiply and divide 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 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.
2𝑥 − 3 = 3 𝑎𝑛𝑑 𝑦 − 4 = 2
𝑥 = 3 𝑎𝑛𝑑 𝑦 = 6
Polynomials5.9 Complex Numbers
Objectives:• Students will add and subtract complex numbers.• Students will multiply and divide 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.
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 − 16𝑖2=
6𝑖 + 12
20=
3
5+
3
10𝑖
Remember, 𝑎 + 𝑏𝑖 is standard form for imaginary numbers.