Download 4.1 Example Guide - Parkway School District

Unit 2 – Function Operations and Equivalence
Standard 4.1 – Add, Subtract, & Multiply Polynomials
This document is meant to be used as an example guide for each of the skills we will be holding
students accountable for with Standard 4.1. This document should be viewed in conjunction with the
4.1 Skill List document.
Skill 4.1 A - Identify like terms. (We cannot add or subtract polynomials if we cannot identify the like
terms.)
1. List the pairs of like terms from the set of given terms. 2, x 2 , 3, 2 x, 0.5 x 2 , − 6 x
Solution:
2, 3
2x, -6x
x2, 0.5x2
The first pair are like terms because they are both constant terms, which means that they are not
attached to any variables. The second pair are like terms because the numbers are attached to the
exact same variable, x. The third pair are like terms because the numbers are attached to the exact
same variable, x2.
2. Identify the pairs of like terms in the polynomial. 2 x 2 + 3 x − 5 + x − 5 x 2 + 14 x 3
Solution:
__2x2, -5x2_
__3x, x___
In the first pair it is important to note that the minus sign in front of the -5x2 term is the same thing as
a negative sign. This is because subtracting a number is the same as adding the opposite of the
number. For example, 1 – 2 is the same as 1 + (-2). We MUST include the sign in front of the term.
Both of the terms in the second pairing are positive numbers as determined by the + signs in front of
their terms in the polynomial. The last thing to note is that there are two other terms that are
singletons, and have no like counterpart.
Skill 4.1 B – Add and subtract like terms.
1. Simplify. 2w − 5 − 3w
Solution: We combine the like terms (2w – 3w) and rewrite the expression as − w − 5 or −5 − w .
2. Simplify. 2 x 2 + 3 x − 5 + x − 5 x 2 + 14 x 3
Solution: We combine the like terms (2x2-5x2) and (3x+x), and then you simplify the expression.
Variable expressions are normally written with the terms in order of decreasing exponents such as
14 x 3 − 3 x 2 + 4 x − 5 . However, the same expression can be rewritten in other ways so it is important to
make sure that the plus (positive) and minus (negative) signs are correct. Students can rearrange
variable expressions as they work them out. One option with specific steps is shown on the next page.
Algebra 1 – Parkway Central High School p.1
Unit 2 – Function Operations and Equivalence
Standard 4.1 – Add, Subtract, & Multiply Polynomials
2 x 2 + 3 x − 5 + x − 5 x 2 + 14 x 3
Original expression
2 x − 5 x + 3 x + x − 5 + 14 x
Group like terms together
2
2
−3 x + 4 x − 5 + 14 x
2
3
3
Simplify like terms
14 x 3 − 3 x 2 + 4 x − 5
Rearragne terms into decreasing power order.
Skill 4.1 C – Add polynomials.
1. Simplify. ( 2 x + 3) + ( −4 x − 7 )
Solution: Adding polynomials is very similar to adding and subtracting like terms. The original
parenthesis used are just to indicate the two separate polynomial expressions. Solve by grouping
together the like term pairs and then simplifying.
( 2 x + 3) + ( −4 x − 7 )
Original expression
2x + 3 − 4x − 7
Rewrite without parenthesis (adding does not change any signs)
2x − 4x + 3 − 7
−2 x − 4
Group like terms together
Simplify like terms
2. Simplify. ( 2 x 2 + x ) + ( −3 x + 8 ) + ( −4 x 2 − 3 x + 1)
Solution: Adding polynomials is very similar to adding and subtracting like terms. The original
parenthesis used are just to indicate the two separate polynomial expressions. Solve by grouping
together the like term pairs and then simplifying.
( 2x
2
+ x ) + ( −3 x + 8 ) + ( −4 x 2 − 3 x + 1)
Original expression
2 x 2 + x − 3x + 8 − 4 x 2 − 3x + 1
Rewrite without parenthesis
2 x 2 − 4 x 2 + x − 3x − 3 x + 8 + 1
Group like terms together
−2 x − 5 x + 9
Simplify like terms
2
Skill 4.1 D –Subtract polynomials.
1. Simplify. ( 2 x − 1) − ( −5 + 4 x )
Solution: There is one very important thing to remember when doing these problems – you must
subtract every term in the polynomial that is being subtracted, in this case, the second polynomial.
There are various ways to attempt this but one of the most basic is to write the subtraction symbol
over and over again as needed.
Algebra 1 – Parkway Central High School p.2
Unit 2 – Function Operations and Equivalence
Standard 4.1 – Add, Subtract, & Multiply Polynomials
( 2 x − 1) − ( −5 + 4 x )
2 x − 1 − −5 − +4 x
Original expression
Rewrite without parenthesis and remember to subtract each term of the
second polynomial (this changes the sign of each term in the second polynomial.)
2x −1 + 5 − 4x
2x − 4x − 1 + 5
−2 x + 4
Group like terms
Simplify like terms
Skill 4.1 E – Multiply by distributing a constant term.
1. Simplify. 2 ( x + 3)
Solution: Use the distributive property to rewrite the expression and then simplify to get 2x + 6.
2 ( x + 3)
Original expression
2⋅ x + 2⋅3
Use the distributive property to multiply each inside term by the outside term
2x + 6
Simplify
2. Simplify. −3 ( 5 x + 1)
Solution: Use the distributive property to rewrite the expression and then simplify to get -15x – 3.
−3 ( 5 x + 1)
−3 ⋅ 5 x + ( −3) ⋅ 1
Original expression
Use the distributive property. The parenthesis is added around the second
negative three to emphasize and help remember its' sign.
−15 x + −3
−15 x − 3
Simplify both multiplication steps
Rewrite the multiple signs (adding a negative is the same as subtracting.)
Skill 4.1 F – Multiply by distributing a variable expression.
1. Simplify x ( 2 x + 3)
Solution: Use the distributive property to rewrite the expression and then simplify to get 2x2 + 3x.
When we multiply x times x, we get x2.
2. Simplify −2 x ( 3 x 2 − 4 x )
Solution: Use the distributive property to rewrite the expression and then simplify to get -6x3 + 8x2.
−2 x ( 3 x 2 − 4 x )
Original equation.
−2 x ⋅ 3 x 2 − 2 x ⋅ −4 x
Distribute the outside term. x times x 2is x 3
−6 x3 + 8 x 2
Simplify.
Algebra 1 – Parkway Central High School p.3
Unit 2 – Function Operations and Equivalence
Standard 4.1 – Add, Subtract, & Multiply Polynomials
Skill 4.1 G – Multiply by distributing a binomial into a binomial.
1. Simplify ( 2 x + 5 )( x − 3) .
Solution:
( 2 x + 5)( x − 3)
2 x ( x − 3) + 5 ( x − 3)
Split up the first term into two parts
2 x − 6 x + 5 x − 15
Distribute both parts
2 x − x − 15
Combine like terms
2
2
Original expression
2. Simplify ( −3 x 2 + 6 x ) ( 4 x − 1)
Solution:
( −3x
2
+ 6 x ) ( 4 x − 1)
Original expression
−3 x 2 ( 4 x − 1) + 6 x ( 4 x − 1)
Split apart the first term
−12 x3 + 3 x 2 + 24 x 2 − 6 x
Distribute both parts
−12 x3 + 27 x 2 − 6 x
Simplify
Skill 4.1 H – Perform multiple operations – distribute and then add or subtract polynomials
1. Simplify 2 ( x + 3) − 4 x ( 5 x − 1) − ( x + 5 )
Solution:
2 ( x + 3) − 4 x ( 5 x − 1) − ( x + 5 )
Original expression
2 x + 6 − 20 x + 4 x − x − 5
Distribute the first and second terms and then
2
subtract both parts of the third paranthesis.
−20 x 2 + 5 x + 1
Combine like terms
Skill 4.1 I – Multiply by distributing a binomial into a trinomial.
1. Simplify ( 3x + 2 ) ( x 2 − 4 x + 6 )
Solution:
( 3x + 2 ) ( x 2 − 4 x + 6 )
3x ( x 2 − 4 x + 6 ) + 2 ( x 2 − 4 x + 6 )
Original expression
Split apart the first paranthesis and distribute
3x 3 − 12 x 2 + 18 x + 2 x 2 − 8 x + 12
3x 3 − 10 x 2 + 10 x + 12
Combine like terms
Algebra 1 – Parkway Central High School p.4
Unit 2 – Function Operations and Equivalence
Standard 4.1 – Add, Subtract, & Multiply Polynomials
Skill 4.1 J – Perform multiple operations – square a binomial
1. Simplify ( 5 x + 2 )
2
Solution:
(5x + 2)
( 5 x + 2 )( 5 x + 2 )
2
Original expression
Rewrite
25 x + 10 x + 10 x + 4
2
Multiply the binomials
25 x + 20 x + 4
2
Skill 4.1 K – Simplify expressions using properties of exponents
1. Simplify
x5
x2
x
2. Simplify  
 y
Solution: When dividing the same base, you subtract the exponents.
3
3. Simplify ( y 3 )
6
3
x
x3
Solution: Each part of the fraction can be rewritten.   = 3
y
 y
Solution: When there are exponents inside and outside a parenthesis, you
multiply the exponents.
4. Simplify 2x 0 y 2
x5
= x3
2
x
(y )
3 6
= y18
Solution: Any non-zero number raised to the power of zero equals one.
2 x 0 y 2 = 2 ⋅1 ⋅ y 2 = 2 y 2
5. Simplify x −2
Solution: A negative exponent is rewritten as a positive exponent by moving the
1
base to the bottom of a fraction. x −2 = 2
x
Practice for the expert level items can be received from the teacher. Students are able to check out a
book for practice if needed.
Algebra 1 – Parkway Central High School p.5