2. Polynomial Operations
In the last section, we introduced polynomial expressions.
#xN ± #xN-1 ± … ± #x ± #
3. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
4. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
5. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
6. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
7. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
8. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
9. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4,
10. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
11. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
12. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient.
13. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient. Hence, 3(5x) = (3*5)x =15x,
14. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
15. Polynomial Operations
In the last section, we introduced polynomial expressions.
Given a polynomial, each monomial is also called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Terms with the same variabe part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x – 3x2 – 5x2 = 9x – 2x2.
Unlike terms may not be combined.
In particular x2 + x2 = 2x2 = x4, and that x2 + x3 = x2 + x3 = x5.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it to the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
When multiplying a number to a polynomial, we may expand
the result using the distributive law: A(B ± C) = AB ± AC.
16. Polynomial Operations
Example D. Expand and simplify.
17. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
18. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12
19. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
20. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
21. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
22. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15
25. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
26. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) =
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
27. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
28. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) =
c. 3x2(2x3 – 4x)
=
29. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x)
=
30. Polynomial Operations
Example D. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
When multiply a term with another term, we multiply the
coefficient with the coefficient and the variable with the
variable.
Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3
c. 3x2(2x3 – 4x) distribute
= 6x5 – 12x3
31. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
32. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
33. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
34. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
35. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
36. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
37. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
38. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
39. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
40. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
41. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ ≡ QP.
42. Polynomial Operations
To multiply two polynomials, we may multiply each term of one
polynomial against other polynomial then expand and simplify.
Example F.
a. (3x + 2)(2x – 1)
= 3x(2x – 1) + 2(2x – 1)
= 6x2 – 3x + 4x – 2
= 6x2 + x – 2
b. (2x – 1)(2x2 + 3x –4)
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
= 4x3 + 4x2 – 11x + 4
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1)
instead, we get the same answers. (Check this.)
Fact. If P and Q are two polynomials then PQ ≡ QP.
A shorter way to multiply is to bypass the 2nd step and use the
general distributive law.
43. Polynomial Operations
General Distributive Rule:
44. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
45. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..
46. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..
47. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
48. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
Polynomial Operations
49. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2
Polynomial Operations
50. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x
Polynomial Operations
51. General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x
Polynomial Operations
52. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12
53. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
54. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
55. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3
56. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2
57. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2 – 2x
58. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2 – 2x – 3x2
59. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2 – 2x – 3x2 + 6x
60. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2 – 2x – 3x2 + 6x + 6
61. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2 – 2x – 3x2 + 6x + 6
= x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-after
we understand more about the multiplication operation.
62. Polynomial Operations
General Distributive Rule:
(A ± B ± C ± ..)(a ± b ± c ..)
= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Example G. Expand
a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify
= x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
= x3 – 2x2 – 2x – 3x2 + 6x + 6
= x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-after
we understand more about the multiplication operation.
63. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
(x – 3)(x2 – 2x – 2) = x3
x3
64. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product,
(x – 3)(x2 – 2x – 2) = x3
65. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2
–3x2
(x – 3)(x2 – 2x – 2) = x3
66. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x)
–3x2
(x – 3)(x2 – 2x – 2) = x3
–2x2
67. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
–3x2
(x – 3)(x2 – 2x – 2) = x3 – 5x2
–2x2
68. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Next collect the x–terms
(x – 3)(x2 – 2x – 2) = x3 – 5x2
69. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x.
–2x
(x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x
6x
70. Polynomial Operations
We include the following method for multiplying polynomials.
Instead of expand all the terms then collect like–terms, we
scan the like–terms and collect them starting from the highest
degree term in the answer. We demonstrate this method and
double check the answer in part b.
Example G. Multiply.
a. (x – 3)(x2 – 2x – 2)
The highest degree term in the product is the x3 term so we
start with the x3–term. It’s x*x2 = 1x3.
Next collect all the x2 terms in the product, they are
–3*x2 and x*(–2x) which gives –3x2 – 2x2 = – 5x2.
Next collect the x–terms and they are x*(–2) + –3*(–2x) = 4x.
The last term is the number term or –3(–2) = 6.
(x – 3)(x2 – 2x – 2) = x3 – 5x2 + 4x + 6 (same as before)
6
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