Practice Algebra Midterm
Introduction to practice algebra midterm:
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.
Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than number.
Example Problems on Practice Algebra Midterm
Practice problem1) Factorize 216p38q3.
Solution for algebra midterm: We can write 216p3 = (6p)3 and 8q3 = (2q)3. So, taking X = 6p and Y = 2q,
we get 216p3 8q3 = (6p)3 (2q)3
X 3 Y 3 = (X Y) (X 2 + XY + Y 2)
= (6p 2q) [(6p)2 +(6p)(2q) +(2q)2]
= (6p 2q)(36p2+12pq+4q2).
Practice problem2) Factorize x2+ 9y2 6xy + 4x 12y + 4.
Solution for algebra midterm: The expression contains a sum x2 + 9y2 + 4. This is a sum of three squares. So, we write x2 + 9y2 6xy + 4x 12y + 4 = x2 + (3y)2 + 22 + 2x (3y) + 2x(2) + 2(2)(3y)
=[x + (3y) + 2]2 = (x 3y + 2)2.
practice problem3) Resolve into factors 4x2 + 12xy + 9y2.
Solutionfor algebra midterm: The given expression can be written as follows:
4x2+ 12xy + 9y2 = (2x)2 + 2(2x)(3y) + (3y)2
Setting X = 2x, Y = 3y, the R.H.S. is X 2 + 2XY + Y 2 and so it is factored
as (X + Y)2. Hence we get 4x2 + 12xy + 9y2 = (2x +3y)2.
practice problem4) Factorize p2 18pq + 81q2.
Solution for algebra midterm: The given polynomial can be written as follows:
p2 18pq + 81q2 = p2 2(p)(9q) + (9q)2
Setting X = p and Y = 9q, the R.H.S. is X 2 2XY + Y 2 and so it is factorised as (X Y)2
Hence we get p2 18pq + 81q2 = (p -9q)2.
Practice Problems on Practice Algebra Midterm
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Algebra midterm question:
1)Factorize 216p38q3.
2)Factorize 8x3 + y 3 + 12x2y + 6xy2.
3) Resolve into factors 8x3 27y3 36x2y + 54xy2.
4)Factorize x2+ 9y2 6xy + 4x 12y + 4.
by: Smith
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