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2.1 Sums and Differences 2.1 Sums and Differences of Polynomialsof Polynomials
GoalsGoals
SWBAT simplify expressions for sums SWBAT simplify expressions for sums and differences of polynomialsand differences of polynomials
SWBAT solve first-degree equations in SWBAT solve first-degree equations in one variableone variable
DefinitionsDefinitions
A A is a numeral, a is a numeral, a variable, or an indicated product of a variable, or an indicated product of a numeral and one or more variables. numeral and one or more variables.
Example: Example:
monomial
232 9,4,,2 yxabm
DefinitionsDefinitions
When looking at the monomialWhen looking at the monomial , the , the number denoted by number denoted by aa is called the is called the
of the variable.of the variable.
The symbol The symbol represents a represents a of of xx, where , where xx is called the is called the and and nn is called the is called the ..
nax
nx
coefficient
power
base
exponent
DefinitionsDefinitions
A monomial with no variable (i.e. 9 or -6) is A monomial with no variable (i.e. 9 or -6) is called a called a ..
The The of a monomial is the of a monomial is the exponent, exponent, nn. If the monomial contains more . If the monomial contains more than one variable, the degree of the monomial than one variable, the degree of the monomial is the is the of the exponents. of the exponents.
Example: What is the degree of Example: What is the degree of ? ? What is the coefficient?What is the coefficient?
constant
degree
428 ba
sum
DefinitionsDefinitions
A monomial or sum of monomials is called a A monomial or sum of monomials is called a . .
The monomials of the expression are called the The monomials of the expression are called the of the polynomial. The of the polynomial. The
coefficients on each term of the polynomial are coefficients on each term of the polynomial are called the coefficients of the polynomial.called the coefficients of the polynomial.
The The of a polynomial is the degree of a polynomial is the degree of the term with the highest degree. of the term with the highest degree.
polynomial
terms
degree
State the coefficients and the degree State the coefficients and the degree of each polynomial.of each polynomial.
1.1.
2.2.
22695 37 xxx
abbaba 933 225
5, -9, 6, -22
Degree: 7
3, -3, -9
Degree: 6
DefinitionsDefinitions Two monomials are said to be Two monomials are said to be
if they have the same variable(s) with if they have the same variable(s) with the same exponent(s) and their only difference is the same exponent(s) and their only difference is their coefficient. their coefficient.
A A is a polynomial with two is a polynomial with two terms. A terms. A is a polynomial with is a polynomial with three terms. three terms. Example:Example: Binomial: Binomial:
Trinomial: Trinomial:
When simplifying a polynomial expression, combine When simplifying a polynomial expression, combine the the terms by adding or subtracting their terms by adding or subtracting their coefficients. coefficients.
liketerms
binomial
trinomial
xx 47 3 1729 3 xx
like
Simplify. Simplify.
3.3.
4. 4.
3564 2 xxx
11310 22 nmmnnm
Given the two polynomials: Given the two polynomials: and and
this is the this is the of the polynomials of the polynomials
this is the this is the of the of the polynomials. polynomials.
xxA 34 2 122 xxB
1234 22 xxxxBA
1234 22 xxxxBA
sum
difference
To simplify these expressions you can To simplify these expressions you can add the like terms. If it is a subtraction add the like terms. If it is a subtraction problem, distribute the negative and then problem, distribute the negative and then combine the like terms. combine the like terms.
Questions 5-8Questions 5-8: Find the sum or : Find the sum or difference and write the answer in difference and write the answer in simplest form.simplest form.
Let Let 975 23 xxA
82 2 xxB
1223 xxC
5.5. AA + + BB
6.6. BB – – CC
7.7. CC – – AA
8.8. AA + + C C - - BB
Questions 9-10Questions 9-10: Simplify.: Simplify.
9.9.
10. 10.
dccdcc 105473 22
5284732 xxxx
2.2 Solving 2.2 Solving EquationsEquations
To solve an equation we canTo solve an equation we can the equation into an the equation into an
equivalent equation to get the solution.equivalent equation to get the solution.transform
Ways to Transform and Ways to Transform and Solve an EquationSolve an Equation::
1. Substituting for either side of the given 1. Substituting for either side of the given equation an expression equivalent to it.equation an expression equivalent to it.
2. Adding to or subtracting from each side 2. Adding to or subtracting from each side of the given equation.of the given equation.
3. Multiplying or dividing each side of the 3. Multiplying or dividing each side of the equation by the same nonzero number.equation by the same nonzero number.*This also includes multiplying by a *This also includes multiplying by a reciprocal*reciprocal*
Make sure when transforming equation to Make sure when transforming equation to only combine only combine terms! terms!like
Solve the equation. Solve the equation.
1.1. 625
m
Solve the equation.Solve the equation.
2. 2. vv 19214
Solve the equation.Solve the equation.
3. 3. 665 xx
Your turn!
Solve #4-6
Solve the equation.Solve the equation.
7. 7. yyy 12223125
Solve the equation.Solve the equation.
8. 8. 6934
552
4
3 bb
Solve for the variable indicated.Solve for the variable indicated.
9. 9. Solve for Solve for nn. .
cdnc 32
15
Solve for the variable indicated.Solve for the variable indicated.
10.10. Solve for Solve for xx
kxhkx 74