Math 1 algebra unit 18
Introduction :
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics
(Source: wikipedia) In this topic we are going to discuss about algebra 1 - unit 18 in math with some example problems.
Algebra 1- Unit 18 example problem in math:
Factorise the expression:
4x2 + y2 + z2 – 4xy – 2yz + 4xz
Solution: We notice that the first three terms are the squares of 2x, y and z respectively. This suggests that we use Identity II, that is,
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Further, the terms with xy and yz are negative in the given expression. This may happen in one of the following two ways :
1. The literal y common to these two terms has a negative coefficient, and x and z have positive coefficients. In this case, the sequence of signs of coefficients of x,y, z is +, –, +.
2. y has a positive coefficient but both x and z have negative coefficients. In this case, the sequence of signs of coefficients of x, y, z is –, +, –.
Since the squares of an expression and its negative are the same, therefore, it does not matter which of the two cases above is taken. Taking the first case, we may write the given expression as follows:
4x2 + y2 + z2 – 4xy – 2yz + 4xz = (2x)2 + (– y2) + z2 + 2 (2x) (– y ) + 2(– y) z + 2(2x) z
we find that a = 2x, b = – y, c = z
Hence, 4x2 + y2 + z2 – 4xy – 2yz + 4xz
= (2x)2 + (– y2) + z2 + 2(2x)(– y) +2(– y)z + 2(2x) z
= {2x + (– y) + z}2
= (2x – y + z)2
= (2x – y + z) (2x – y + z)
Algebra 1- Unit 18 example problem in math:
Factorise the expression 8x3 + 27y3 + 36x2y + 54xy2.
Solution: We notice that the first two terms are the cubes of 2x and 3y respectively. Also, the remaining two terms have a factor 3.that is,
(a + b)3 = a3 + b3 +3ab (a + b)
We may write the given expression as follows:
8x3 + 27y3 + 36x2y + 54xy2 = (2x)3 + (3y)3 + 3(2x)(3y)(2x + 3y)
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics
(Source: wikipedia) In this topic we are going to discuss about algebra 1 - unit 18 in math with some example problems.
Algebra 1- Unit 18 example problem in math:
Factorise the expression:
4x2 + y2 + z2 – 4xy – 2yz + 4xz
Solution: We notice that the first three terms are the squares of 2x, y and z respectively. This suggests that we use Identity II, that is,
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Further, the terms with xy and yz are negative in the given expression. This may happen in one of the following two ways :
1. The literal y common to these two terms has a negative coefficient, and x and z have positive coefficients. In this case, the sequence of signs of coefficients of x,y, z is +, –, +.
2. y has a positive coefficient but both x and z have negative coefficients. In this case, the sequence of signs of coefficients of x, y, z is –, +, –.
Since the squares of an expression and its negative are the same, therefore, it does not matter which of the two cases above is taken. Taking the first case, we may write the given expression as follows:
4x2 + y2 + z2 – 4xy – 2yz + 4xz = (2x)2 + (– y2) + z2 + 2 (2x) (– y ) + 2(– y) z + 2(2x) z
we find that a = 2x, b = – y, c = z
Hence, 4x2 + y2 + z2 – 4xy – 2yz + 4xz
= (2x)2 + (– y2) + z2 + 2(2x)(– y) +2(– y)z + 2(2x) z
= {2x + (– y) + z}2
= (2x – y + z)2
= (2x – y + z) (2x – y + z)
Algebra 1- Unit 18 example problem in math:
Factorise the expression 8x3 + 27y3 + 36x2y + 54xy2.
Solution: We notice that the first two terms are the cubes of 2x and 3y respectively. Also, the remaining two terms have a factor 3.that is,
(a + b)3 = a3 + b3 +3ab (a + b)
We may write the given expression as follows:
8x3 + 27y3 + 36x2y + 54xy2 = (2x)3 + (3y)3 + 3(2x)(3y)(2x + 3y)