Factorization synchronous exercise (problem solving)
answer the question
9. Break down the following categories:
①a2+ 10a+25②m2- 12mn+36 N2
③xy3-2x2y 2+x3y④(x2+4 y2)2- 16x2y 2
10. Given x=- 19 and y= 12, find the value of the algebraic expression 4x2+ 12xy+9y2.
1 1. It is known that │x-y+ 1│ and x2+8x+ 16 are reciprocal. Find the value of x2+2xy+y2.
Answer:
9.①(a+5)2； ②(m-6n)2； ③xy(x-y)2； ④(x+2y)2(x-2y)2
Through the above study on the purpose of factorization synchronous exercise, I believe the students have mastered it very well, and I wish them good results in the exam.
Factorization synchronous exercise (fill in the blanks)
Are the students familiar with the content of factorization? Now students need to do the following exercises.
Factorization synchronous exercise (fill in the blanks)
fill-in-the-blank question
5. It is known that 9x2-6xy+k is completely flat, so the value of k is _ _ _ _ _.
6.9a2+(________)+25b2=(3a-5b)2
7.-4x2+4xy+(_______)=-(_______)。
8. Given a2+ 14a+49=25, the value of A is _ _ _ _ _ _ _.
Answer:
5.26。 -30ab 7。 -y2； 2x-Y 8。 -2 or-12
Through the above study on the purpose of factorization synchronous exercise, I believe the students have mastered it very well, and I wish them good results in the exam.
Factorization synchronous exercise (multiple choice questions)
Students study hard. The following is the factorization synchronization exercise provided by the teacher.
Factorization synchronous exercise (multiple choice questions)
Multiple choice
1. Given that y2+my+ 16 is completely flat, the value of m is ().
A8 b . 4 c . 8d . 4
2. The following polynomials can be factorized by the complete square formula ().
a . x2-6x-9 b . a2- 16a+32 c . x2-2xy+4 y2 d . 4a 2-4a+ 1
3. All the following belong to the correct decomposition factor is ()
a . 1+4 x2 =( 1+2x)2 b . 6a-9-a2 =-(a-3)2
c . 1+4m-4 m2 =( 1-2m)2d . x2+xy+y2 =(x+y)2
4. Factorizing x4-2x2y2+y4, the result is ()
A.(x-y)4b .(x2-y2)4c .[(x+y)(x-y)]2d .(x+y)2(x-y)2
Answer:
1.C 2。 D 3。 B 4。 D
I believe that the students have completed the knowledge exercise of the above factorization synchronous exercise (multiple choice), and I hope that the students can do well in the exam.
Algebraic expression factorization unit test paper (fill in the blank)
The following is the practice of filling in the blanks in the test paper of algebraic expression multiplication, division and factorization unit. I hope the students can finish it well.
Fill in the blanks (4 points for each small question, ***28 points)
7.(4 points) (1) When x _ _ _ _ _ _ _ _ _ x-4) 0 =1; (2)(2/3)2002×( 1.5)2003÷(﹣ 1)2004= _________
8.(4 points) Decomposition factor: A2-1+B2-2ab = _ _ _ _ _.
9.(4 points) (Wanzhou District, 2004) As shown in the figure, box this box with the length, width and height of x, y and z respectively. As shown in the figure, the length of the packing belt should be at least _ _ _ _ _ _ _ (unit: mm) (expressed by an algebraic expression containing x, y and z).
10.(4 points) (Zhengzhou, 2004) If (2a+2b+1) = 63, then the value of a+b is _ _ _ _ _ _ _.
1 1.(4 points) (Changsha, 2002) The figure shows Yang Hui's triangle table, which can help us write the coefficients of the expansion of (a+b)n (where n is a positive integer) according to the law. Please carefully observe the rules in the table and fill in the missing coefficients in the (a+b)4 expansion.
(a+b) 1 = a+b；
(a+b)2 = a2+2ab+B2；
(a+b)3 = a3+3a2b+3ab 2+B3；
(a+b)4 = a4+_ _ _ _ _ _ _ _ _ _ _ _ a3 b+ _ _ _ _ _ _ a2 B2+_ _ _ _ _ _ ab3+B4。
12.(4 points) (Jingmen, 2004) The germination of some plants follows a law: the new buds produced in that year do not germinate the next year, and the old buds germinate every year thereafter. See the table below for the germination rule (let the number of new buds before the first year be a).
Year n,12345 ...
Old bud rate aa2a3a5a…
Buding rate 0 A 2 A 3 A …
Total budding rate a2a3a5a8a…
At this rate, the ratio of the number of old buds to the total number of buds in the eighth year is _ _ _ _ _ (accurate to 0.00 1).
13.(4 points) If the value of a holds that x2+4x+a = (x+2) 2- 1, then the value of a is _ _ _ _ _ _ _.
Answer:
7.
Test site: zero exponential power; Power of rational number. 1923992
Special topic: calculation problems.
Analysis: (1) According to the meaning of zero index, we can know that X ≠ 4 ≠ 0 is x ≠ 4;
(2) Calculate according to the power operation rule and rational number operation order.
Solution: Solution: (1) According to the meaning of zero exponent, we can know that x﹣4≠0,
x≠4；
(2)(2/3)2002×( 1.5)2003÷(﹣ 1)2004=(2/3×3/2)2002× 1.5÷ 1= 1.5.
Comments: The main knowledge points are: the operation of zero exponential power, negative exponential power and square. The negative exponent is the reciprocal of the positive exponent, and the power of any non-zero number is equal to 1.
8.
Test center: factorization-grouping decomposition method. 1923992
Analysis: When the decomposition formula is four terms, the group decomposition method should be considered. A2+B2-2ab in this problem just conforms to the complete square formula and should be counted as a group.
Solution: Solution: A2- 1+B2-2AB.
=(a2+b2﹣2ab)﹣ 1
=(a﹣b)2﹣ 1
=(a﹣b+ 1)(a﹣b﹣ 1).
So the answer is: (A-B+ 1) (A-B- 1).
Comments: This topic examines the factorization of grouping decomposition method. The difficulty is whether to use pairwise grouping or trinity grouping. After grouping, the next step can be decomposed.
9.
Test center: column algebra. 1923992
Analysis: We mainly study reading pictures, and draw a conclusion by using the information in the pictures: the length of the belt is divided into three parts: the belt is equal to the length with 2 segments, denoted by 2x, the belt is equal to the width with 4 segments, denoted by 4y, and the belt is equal to the height with 6 segments, denoted by 6z, so the total length is the sum of these three parts.
Solution: If the band is equal to length, the total length is 2x+4y+6z; if the band is equal to width, the total length is 4y; if the band is equal to height, the total length is 6z.
Comments: The key to solving the problem is to read the meaning of the question and find the equivalent relationship of the required quantity.
10.
Test center: variance formula. 1923992
Analysis: regard 2a+2b as a whole, calculate the value of 2a+2b by square difference formula, and further calculate the value of (a+b).
Solution: ∫ (2a+2b+1) (2a+2b-1) = 63,
∴(2a+2b)2﹣ 12=63，
∴(2a+2b)2=64，
2a+2b= 8，
If both sides are divided by 2 at the same time, a+b = 4.
Comments: This question examines the square difference formula, and the application of the overall idea is the key to solving the problem. Students are required to answer carefully and look at it as a whole (2a+2b).
1 1
Test center: complete square formula. 1923992
Special topic: ordinary type.
Analysis: observe the law of this problem. The data in the next row is the sum of two adjacent numbers in the previous row. Just fill it out according to the law.
Solution: solution: (a+b) 4 = A4+4a3b+6a2b2+4ab3+B4.
Comments: On the premise of investigating the complete square formula, I have a deeper understanding of Yang Hui Triangle.
12
Test center: regular type: types of numbers. 1923992
Topic: Chart types.
Analysis: According to the data in the table, it is found that the number of old buds is always the sum of the first two numbers, the number of new buds is the corresponding number of old buds in the previous year, and the total number of buds is equal to the sum of the corresponding number of new buds and old buds. According to this rule, the number of old buds in the eighth year is 2 1a, the number of new buds is 13a, and the total number of buds is 34a, so the ratio is
2 1/34≈0.6 18.
Solution: As can be seen from the table, the number of old buds is always the sum of the first two numbers, the number of new buds is the corresponding number of old buds in the previous year, and the total number of buds is equal to the sum of the corresponding number of new buds and old buds.
So in the eighth year, the number of old buds is 2 1a, the number of new buds is 13a, and the total number of buds is 34a.
Then the ratio is 2 1/34 ≈ 0.6 18.
Comments: According to the data in the table, find out the law of the number of new buds and old buds, and then solve it. The key rule of this problem is that the number of old buds is always the sum of the first two numbers, the number of new buds is the corresponding number of old buds in the previous year, and the total number of buds is equal to the sum of the corresponding number of new buds and old buds.
13.
Test site: mixed operation of algebraic expressions. 1923992
Analysis: Calculate the right side of the equation with the complete square formula, and then list and solve the equation according to the equality of constant terms.
Answer: Solution: ∫ (x+2) 2-1= x2+4x+4-1,
∴a=4﹣ 1，
The solution is a = 3.
So the answer to this question is: 3.
Comments: This question examines the complete square formula, memorizing the formula, and equating the formula according to the constant term is the key to solving the problem.
Students have mastered the practice of algebraic multiplication and division and factorization unit test papers. I hope students can make good reference to meet the exam work.
Algebraic expression factorization unit examination paper (multiple choice question)
The following is the practice of multiplication, division and factorization of algebraic expressions for multiple-choice questions in the unit test paper. I hope the students can finish it well.
Algebraic expression factorization unit test paper
Multiple choice questions (4 points for each small question, ***24 points)
1.(4 points) The following calculation is correct ()
A.a2+b3=2a5B.a4÷a=a4C.a2a3=a6D。 (﹣a2)3=﹣a6
2.(4 points) (x﹣a)(x2+ax+a2)' The calculation result is ().
a.x3+2ax+a3b.x3﹣a3c.x3+2a2x+a3d.x2+2ax2+a3
3.(4 points) The following is taken from a classmate's calculation in an exam:
①3x3(﹣2x2)=﹣6x5 ②4a3b÷(﹣2a2b)=﹣2a ③(a3)2=a5④(﹣a)3÷(﹣a)=﹣a2
The correct number is ()
1。
4.(4 points) If x2 is the square of a positive integer, then the square of the integer after it should be ().
a.x2+ 1b.x+ 1c.x2+2x+ 1d.x2﹣2x+ 1
5.(4 points) The following decomposition factors are correct ()
a.x3﹣x=x(x2﹣ 1)b.m2+m﹣6=(m+3)(m﹣2)c.(a+4)(a﹣4)=a2﹣ 16d.x2+y2=(x+y)(x﹣y)
6.(4 points) (Changzhou, 2003) As shown in the figure: In the rectangular garden ABCD, AB=a, AD=b, there is a rectangular road LMPQ and a parallelogram road RSTK. In the garden. If LM=RS=c, the area of the green part in the garden is ().
a.bc﹣ab+ac+b2b.a2+ab+bc﹣acc.ab﹣bc﹣ac+c2d.b2﹣bc+a2﹣ab
Answer:
1, test site: division in the same base power; Merge similar projects; Multiplication with the same base; Power and products. 1923992
Analysis: according to the division of the same base number, subtract the index of the same base number; Same base powers multiplication, exponential addition of base constant; Multiply the power by the exponent of the same base number, and each option is calculated and solved by exclusion method.
Solution: Solution: A, a2 and b3 are not similar items and cannot be merged, so this option is wrong;
B, it should be a4÷a=a3, so this option is wrong;
C, should be a3a2=a5, so this option is wrong;
D, (-A2) 3 =-A6, correct.
So choose D.
Comments: This topic examines the properties of merging similar terms, division of same base powers, multiplication of same base powers and idempotency of power. Mastering the essence of operation is the key to solving the problem.
2.
Test center: polynomial multiplication polynomial. 1923992
Analysis: According to the rule of polynomial multiplication, multiply each term of one polynomial with each term of another polynomial, and then add the products to calculate.
Solution: solution: (x-a) (x2+ax+a2),
=x3+ax2+a2x﹣ax2﹣a2x﹣a3，
=x3﹣a3.
So choose B.
Comments: This question examines the law of polynomial multiplication. When merging similar items, pay attention to whether the indexes and letters in the items are the same.
3.
Test center: single item multiplied by single item; The power of power and the power of products; The division of power with the same base; Division of algebraic expressions. 1923992
Analysis: According to the law of single item multiplication, the law of single item division, the nature of power and the division of same base powers, the options are calculated and solved by exclusion method.
Solution: Solution: ①3x3(﹣2x2)=﹣6x5, correct;
② 4a3b ÷ (-2a2b) =-2a, correct;
③ It should be (a3)2=a6, so this option is wrong;
④ It should be (﹣ a) 3 ﹣ (﹣ a) = (﹣ a) 2 = A2, so this option is wrong.
So ① ② Two items are correct.
So choose B.
Comments: This topic examines the multiplication of monomial and monomial, the division of monomial and monomial, the power of power, and the division of the same base power. Pay attention to master all the algorithms.
four
Test center: complete square formula. 1923992
Special topic: calculation problems.
Analysis: first find the following integer x+ 1, and then solve it according to the complete square formula.
Solution: x2 is the square of a positive integer, followed by x+ 1.
The square of an integer after it is: (x+ 1) 2 = x2+2x+ 1.
So choose C.
Comments: This question mainly examines the complete square formula, and memorizing the formula structure is the key to solving the problem. Complete square formula: (ab) 2 = A2AB+B2.
5,
Test sites: factorization-cross multiplication, etc. The significance of factorization. 1923992
Analysis: According to the definition of factorization, a polynomial is transformed into the product of several algebraic expressions. The deformation of this formula is called factorization of this single item, and the result of factorization should be correct.
Solution: A, x3-x = x (x2-1) = x (x+1) (x-1), and the decomposition is incomplete, so this option is wrong;
B, decompose m2+m﹣6=(m+3)(m﹣2) by cross multiplication, which is correct;
C, algebraic multiplication, not factorization, so this option is wrong;
D. Without the sum of squares formula, x2+y2 cannot decompose the factor, so this option is wrong.
So choose B.
Comments: This question examines the definitions of factorization and cross factorization. Note: (1) factorization is a polynomial, and the result of factorization is a product. (2) Factorization must be thorough until it can no longer be decomposed.
six
Test sites: factorization-cross multiplication, etc. The significance of factorization. 1923992
Analysis: According to the definition of factorization, a polynomial is transformed into the product of several algebraic expressions. The deformation of this formula is called factorization of this single item, and the result of factorization should be correct.
Solution: A, x3-x = x (x2-1) = x (x+1) (x-1), and the decomposition is incomplete, so this option is wrong;
B, decompose m2+m﹣6=(m+3)(m﹣2) by cross multiplication, which is correct;
C, algebraic multiplication, not factorization, so this option is wrong;
D. Without the sum of squares formula, x2+y2 cannot decompose the factor, so this option is wrong.
So choose B.
Comments: This question examines the definitions of factorization and cross factorization. Note: (1) factorization is a polynomial, and the result of factorization is a product. (2) Factorization must be thorough until it can no longer be decomposed.
6.
Test center: column algebra. 1923992
Special topic: application problem.
Analysis: the area of the green part =S rectangle ABCD-s rectangle LMPQ-s? The overlapping part of RSTK+.
Solution: ∫ The area of rectangle is ab, the area of rectangular road LMPQ is bc, the area of parallelogram road RSTK is ac, and the overlapping area of rectangle and parallelogram is C2.
The area of the green part is AB-BC-AC+C2.
So choose C.
Comments: It should be noted that the overlapping part of the pavement is a parallelogram with an area of c2.
When using letters to represent numbers, pay attention to writing:
① Multiplication symbols in algebraic expressions are usually abbreviated as ""or omitted, and numbers are usually multiplied by numbers with "×";
(2) When there is a division operation in the algebraic expression, it is generally written in fractional form;
③ Numbers are generally written in front of letters;
Those with scores should be written as fake scores.
Students have mastered the practice of algebraic multiplication and division and factorization unit test papers. I hope students can make good reference to meet the exam work.