As we go deeper to the study of College Algebra, we should first familiarize ourselves to the different symbols and terms used in solving Algebra.
First, we should have the following symbols for groupings. The first three (3) symbols are used for multiplication while the fourth (4) symbol is used for division.
1. ( )- parentheses
2. [ ]- brackets
3. { }- braces
4. — - vinculum
As we proceed with our discussion, we will be able to encounter such symbols. In performing problems or mathematical expressions with these symbols, we should first solve equations in the parentheses; second, in the
Examples: Simplify the following by removing the given symbols.
1. 8[6-5(4+1)] = 8[6-5(5)] = 8(6-25) = 8(-19) = -152
2. -2[-5+2(1-5)] = -2[-5+2(-4)] = -2(-5-8) = -2(-13) = 26
3. -6+5{8-4[6+5(5-3)]} = -6+5{8-4[6+5(2)]} = -6+5[8-4(6+10)] = -6+5[8-4(16] = -6+5(8-64) = -6+5(-56) = -6-280 = -283
NOTE: We should recall from our past lessons in Mathematics when we were still in high school that multiplication and division dominates addition and subtraction. That is, in example 1) 8[6-5(4+1)], we can! ’t just directly multiply 8 to 6 then subtract 5 then mu! ltiply b y 4 and add 1. If we will follow this procedure, it will give 173 as our final answer and obviously it is wrong because it is not the same with the correct answer previously shown.
Another is when adding or subtracting 2 or more numbers with different signs, for example a) 5-3; b) -5+3. In solving this, we should first identify that this two problems have unlike signs. In the first example, 5 is positive while 3 is negative. In the second example, 5 is negative and 3 is positive. If given unlike signs, we should use the operation subtraction. Therefore, we should subtract the bigger number from the smaller number then, copy the sign of the bigger number. In example a) the final answer is +2 or 2, while in b) the final answer is -2. It is negative because we copied the sign of the bigger number 5.
! While in adding and subtracting numbers with like signs, we should use the operation addition then copy the sign. For example a) 5+3+2 = 10; b) -5-3-2 = -10. In example a), we simply add 5, 3 & 2 then we copied the positive sign since all of them are positive. In example b), since they all got negative signs, therefore we should add 5, 3 &2 then copy the sign which is negative.
Algebraic Terms
An algebraic term is a mathematical notation composed of numbers and letters that represent a single quantity or number in a polynomial. For example, -5; x; 4x; -2xy. Each of them is an algebraic term. A! n algebraic term maybe in the form of a m onomial, a polynomial with only one term; binomial, polynomial with two terms; trinomial, polynomial with three terms; and a polynomial, composed of one or more monomials.
Now, let’s identify the differents parts of an algebaric term. For example, 4x³. 4 is called the numerical coefficient, x is called the literal coefficient, and 3 is called the exponent/degree/power.
In solving algebraic expressions, we should able to differentiate like terms/similar terms from! unlike terms/dissimilar terms. Like terms have the same literal coefficients and exponents. For example, 3x²y³, -y³x² and 100y³x². These are all similar terms because they have the same exponents for x and y as 2 and 3 respectively. While unlike terms don’t have the same literal coefficients and exponents. For example, -3x² y, yx³, and 87x³y² z. These are all dissimilar terms because they have different exponents for x, y and other terms don’t have a literal coefficient z.
Addition and Subtraction of Polynomials
In addition and subtraction of polynomials, it’s just the same with adding or subtracting real numbers. Only that we should only add or subtract l! ike terms (depends on the operation given). For example,
4x² +5x-5 added with 6x² -3x+4
4x² +5x-5
+ 6x² -3x+4
-------------
10x² +2x-1
2. -a³+a² -3a+9 subtracted with -4a ³ +a² - 5a+6
-a³ +a² -3a+9
− -4a³ +a² -5a+6
-----------------
↓
-a³+a² -3a+9
+ 4a³ -a² +5a-6
-----------------
3a³ +2a+3
NOTE: In adding and subtracting polynomials, we should align like terms in a single column then perform the operation. In subtracting polynomials, we should change all the sign of the terms in the subtrahend, then perform the operation with the new sign in the subtrahend.
In multiplying and dividing n! umbers/mathematical expressions with like signs, we should perform the indicated operation, that's either multiplication or division, then make the sign of the final answer positive. For example,
- (3)(15) = +45 or 45
(-2)(-8) = +16 or 16
3. -18 ÷ -6 = +3 or 3
While in multiplying and dividing numbers/mathematical expressions with unlike signs, we again should perform the indicated operation, then make the sign of the final answer as negative. For example,
- -100 ÷ 2 = -50
- 85 ÷ -5 = -17
- 15 * -2 = -30
Table 1.0 Summary of the sign used in Multiplication and Division of Polynomials
(+)(+) = +
(+)(-) = -
(-)(-) = +
(-)(+) = -
Therefore, if two terms have like signs the answer is always positive, while if two terms have unlike signs, the answer is negative.
Multiplication and Division of Polynomials
In multiplication and division of polynomials, we should learn the rule in index laws.
Parts: x² --> x is called the base, while 2 is the exponent/power/degree
Note: To show an exponent, we shall use a^. So a^3 means a³ .
- (a^m)(a^n) = a^m+n ; example: (x^3)(x^2) = x^5
- (a^m)^n = a^mn ; example: (b^3)^2 = b^(3)(2) = b^6
- (ab)^m = a^m●b^m ; example: (2xy)^2 = (2)^2 ●(x)^2 ●(y)^2 = 4x^2 y^2
Note: (a+b)^m ≠ a^m + b^m
4. (a/b)^m = a^m/b^m iff b≠0; example: (x/y)^2 = x^2/y^2
NOTE: Let x be any number . x/0 = α ->means undefined. 0/x = 0. 0/0 = undetermined
5. a^m/a^n = a^m-n iff m>n; example: x^3/x = x^3-! 1 = x^2
a^m/a^n = 1/a^n-m iff n>m; example: a^6/a^9 = 1/a^9-6 = 1/a^3
a^m/a^n = 1 iff m=n; example: b^3/b^3 = 1
NOTE: Any value raise to zero is one. Example: 0º = 1; xº = 1; 1000º = 1
6. a^-n = 1/a^n; example: x^-5 = 1/x^5
7. 1/a^-n = a^n; example: 1/x^-3 = x^3
Sample Test I. Add the three expressions in each of the problems in 1-5.
1) 3x+2y-4z
-3x - y + z
5x+8y-2z
-------------
2) 7ab - 6bc + 5ac
-ab +8cb -6ac
13ab -cb +ac
-----------------
3) -9pd -dq -9pq
6pd+8dq+3pq
-dq -pq
------------------
4) 7am-2ap+5pd
-ma -8pd
3pa+6odp
-----------------
5)45qc-15a+5
-5cq-15a+10
6qc +a -1
----------------
Key Answers:
- 5x + 9y - 5z
- 19ab + cb
- -3pd + 6dq - 7pq
- 6am + pa + 57dp
- 46qc - 29a + 14
Sample Test II. Answer the following problems.
1) 6a-b+3c
− -8a-2b+c
-------------
2) 5xy-z+2ab
− -10yx+3z+ab
-----------------
3) 5pq-z+x²
− -3qp+2z+3x
----------------
4) 6pq+q² -qp²
− -16q² -8qp+qp²
------------------
5)16vz-1+x
− -2x-12zv
------------
Key Answers:
- 14a + b + 2c
- 15xy - 4z + ab
- 8pq - 3z + x² - 3x
- 14qp + 17q² - 2qp²
- 28vz - 1 + 3x
Sample Test III. Simplify the following.
- (4x^4)(6x^6) =
- (5a^-5)(a^3) =
- 6x^5∙y^6∙ z^7 ÷ 36x^2∙ y^7∙ z^9 =
- a^-3∙ b^5∙ c^-7 ÷ a^-6∙ b^5∙ c^-4 =
- -2(4x² +3x-1) =
- -5x² (8x³ +4x² -6x+5) =
- (3x+4)(x+5) =
- 15x² -12x+6 ÷ 3 =
- 5m³ -9m²+10m ÷ 5m =
- 2p³ +9p² +27 ÷ 2p-3 =
- 4y³ -12y² +5y ÷ 4y =
- 36x³ y-40x² y² +21xy³ ÷ 6xy²
Key Answers with Explanations:
- 24x^10 ->rule no. 1 of index laws
- 5a^-2 or 5/a^2 ->rule no. 1 of index laws; no. 6 index laws
- x³ /6yz² ->rule no. 5 of index laws
- a ^3 c^-3 or a^3/c^3 ->rule no. 5, 6 and 7 of index laws
- -8x² -6x+2 -> since the 3 terms in the parentheses are unlike terms, therefore we can't add or subtract them. To solve this, we will just distribute -2 with the 3 terms in the parentheses by multiplying each terms with -2.
- -40x^5-20x^4+30x^3-25x^2 -> (the same with no. 5)
- 3x² +19x+20 -> since we are to multiply (3x+4) with (x+5), we use the FOIL method. Foile method means multiplying the (F) first terms 3x & x, (O)outer terms 3x & 5, (I) inner terms 4 & x, and (L) last terms 4 ! & 5.
- 5x² -4x+2 ->we just divided 15x² -12x+6 by 3
- m² - 9m÷ 5 +2 -> (the same with no. 8) then use rule no. 5 of index laws
- p^6 + 9p^5/2+27/2p^-3 -> we will just divide the numerator by the denominator (the same with no. 9) then use the index laws
- y ² -3y+5/4 ->(the same with no. 10) then use the rule no. 5 in index laws
- 6x ² /y-20x/3+7y/2 ->(the same with no.11) then use rule no. 5 in index laws
add and subtract polynomials
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