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Factorization of equation x ^ 3 + x ^ 2-8x-8 = 0
x^3+x^2-8x-8=0
x^2(x+1)-8(x+1)=0
(x+1)(x^2-8)=0
(x+1)(x+2√2)(x-2√2)=0
The solution is x = - 1 or x = - 2 √ 2 or x = 2 √ 2
x^2(x+1)-8(x+1)=0
(x+1)(x^2-8)=0
x=-1, 2√2, -2√2
The cubic equation of one variable cannot be solved by factorization
Some questions about factorization by formula method! Not about simple factorization of roots=
（1）、 3(x+1)(x-1)-(3x+2)(2-3x) (2)、 (x+y)²-x² （3）、 （3a+2b）(3a-2b) (4)、 （x+1/2）(x²+1/4)(1/2-x)
(1) = 3 (x2-1) - (6x + 4 - 9x2-6x) = 3 x2-3-4 + 9 x2 = 12 X2 (the square after X cannot be displayed) - 7
（2）=X2+2XY+Y2-X2=Y2+2XY
（3）=9a2-4b2
（4）=-（X2-1/4)(x2+1/4)=-(x4-1/16)=-X4+1/16
Calculation by factorization
(1) - one quarter
(1-one hundredth)
The original formula = (1-1 / 2 square) (1-1 / 3 square) (1-1 / 4 square); (1 / 10 square) = (1-1 / 2) (1 + 1 / 2) (1-1 / 3) (1 + 1 / 3) (1-1 / 4) (1 + 1 / 4); (1-1 / 10) (1 + 1 / 10) = (1 / 2) * (3 / 2) * (2 / 3) (4 / 3) * (3 / 4) * (5 / 4); (9 / 10) * (11 / 10) all the multipliers in the middle are reduced to the first and last two terms = (1
The original formula = (1-1 / 2 square) (1-1 / 3 square) (1-1 / 4 square)... (1 / 10 square)
=（1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)....(1-1/10)(1+1/10)
=(1/2)*(3/2)*(2/3)(4/3)*(3/4)*(5/4)....(9/10)*(11/10)
=(1/2)*(11/10)
=11/20
The application formula method in factorization,
(x + a) (x + b) = 0. The value of X is - A or - B
The left side is equal to the multiplication of polynomials, and the right side is equal to 0
It's better to be ternary or multivariate, but it's hard,
Three polynomials are given: 1 / 2x & # 178; - 2x, 1 / 2x & # 178; + 1,1 / 2x & # 178; + 2x-1. Two of them are selected to carry out the addition operation
Factorization of results
1/2x²-2x+1/2x²+1
=x²-2x+1
=(x-1)²
(1/2x²-2x)+(1/2x²+2x-1)
=x²-1
=(x+1)(x-1)
(1/2x²+1)+(1/2x²+2x-1)
=x²+2x
=x(x+2)
On the relationship between the root and coefficient of quadratic equation with one variable
Given that a ≠ B, and a * a + 3a-7 = 0, b * B + 3b-7 = 0, find the value of 1 / A + 1 / b
Let me tell you two formulas that can be used when ∈ is greater than 0
X1+X2= -b/a
X1*X2=c/a
Then a and B of the above two questions are the two roots of X * x + 3x-7 = 0, so it can be changed into B / AB (1 / a) + A / AB (1 / b), so after the combination, it is a + B / AB, using the Vader theorem
Don't forget to give me points
3/7
Second grade factorization exercises become too urgent and difficult
(65+2)^2*(65-2)^2=260
(^ 2 is the square)
=(65+2）×（65-2)×（65+2）×(65-2)=(65^2-2^2)×(65^2-2^2)=(65^2-2^2)^2
（65+2）×（65-2)×（65+2）X(65-2)
Three polynomials are given: 1 / 3 x ^ 2 + 4x-1, 4 / 3 x ^ 2 + 2x + 1, 1 / 3 x ^ 2-3x + 6. Please select your favorite polynomials for subtraction and find them
The value of the difference between two polynomials when x = 2
(1 / 3 x ^ 2 + 4x-1) - (1 / 3 x ^ 2-3x + 6)
=(x^2+4x-1-x^2+3x-6)/3
=(7x-7)/3
When x = 2jf
The above formula = (7 * 2-7) / 3 = 7 / 3: - P
(1 / 3 x ^ 2 + 4x-1) - (1 / 3 x ^ 2-3x + 6)
=(x^2+4x-1-x^2+3x-6)/3
=(7x-7)/3
When x = 2jf
The above formula = (7 * 2-7) / 3 = 7 / 3
Do not understand... Can you type mathematical symbols? There are a lot of incorrect symbols in it...
10. Given that one root of the quadratic equation AX ^ 2 + BX + C = 0 is 1, and a and B satisfy the equation B = √ (A-2) + √ (2-a-3), find the root of equation 1 / 4Y ^ 2-C = 0
(please write the process)
The - 3 of (2-a-3) should be outside the root
Because B = √ (A-2) + √ (2-A) - 3
A-2 and 2-A are opposite numbers
So A-2 = 2-A = 0
So a = 2
So B = - 3
If ax ^ 2 + BX + C = 0
2-3+c=0
C=1
If 1 / 4Y ^ 2-C = 0
1/4y^2=1
y=±2
B = √ (A-2) + √ (2-a-3) is there a problem?
And where did the last y come from?
top
Ask 200 questions about factorization and integral multiplication and division in the second grade of junior high school
Is on the second day, at least 200, there must be an answer which ~!
Chapter 15 multiplication, division and factorization of integers
Section 4 factorization
The first lesson raises the common factor method
Follow up training:
1. The deformation of the following formulas from left to right belongs to factorization ()
A． B．
C． D．
2. Observe the following formulas: 1; 2; 3; 4; 5; 6. Among them ()
A．①②⑤ B．②④⑤ C．②④⑥ D．①②⑤⑥
3. When factoring a polynomial, the common factor to be extracted is ()
A．3mn B． C． D．
4. The following factorizations are correct
①4a－a3b2＝a（4－a2b2）；②x2y－2xy＋xy＝xy（x－2）；③－a＋ab－ac＝－a（a－b－c）；④9abc－6a2b＝3abc（3－2a）；⑤ x2y＋ xy2＝ xy（x＋y）
A. 0 B.1 C.2 d.5
5. If, then a is ()
A． B． C． D．
6. Decompose a polynomial (n is a positive integer greater than 2) into ()
A． B． C． D．
7. The result of factoring a polynomial is ()
A． B． C． D．
8. Convert a polynomial into several integers_______ It's called factoring this polynomial
9. Using factorization to calculate 32 × 3.14 + 5.4 × 31.4 + 0.14 × 314 =________ .
10. Write the common factors of the following polynomials respectively:
（1） ： ；
（2） ： ；
（3） ： ；
（4） ： ；
11. Given a + B = 13, ab = 40, the result is______________ .
12. The following formulas are decomposed by the method of quoting the common factor
（1） （2）
13. When x = 2, y = 1, find the value of the algebraic expression
15.4 answers for session 1:
1. D (dial: to judge whether factorization must satisfy two points: one is that the left side of the equation is a polynomial, the other is the form of integral product of the equation)
2. D (dial: to see if the common factor method can be used, the key to factorization is whether there is a common factor in the polynomial) 3. B (dial: the coefficient of the common factor takes the greatest common divisor of each coefficient, and the same letter takes the lowest exponential power to ensure that the first term of the extracted polynomial is positive)
4. B (dialing: ① correct; ② missing items after extracting the common factor; ③ the last item after extracting the common factor should be + C; ④ the common factor should be 3AB; ⑤, ⑥)
5. D (dial: divide by)
6. D (dial: the common factor is the lowest power of the same letter, then divide by the common factor)
7. C: the common factor of this question is, we must mention it completely
8. Product
9、314
10、（1） ；（2） ；（3） ；（4）
11、520
12. (1) original =; (2) original =;
13、
=
＝ ＝x（x＋y）
Substituting x = 2, y = in, the original formula is 2 × (2 +) = 5
Formula method in the second lesson (1)
Follow up training:
1. Among the following formulas, the one that can't be decomposed by the square difference formula is ()
A． B． C．49 D．
2. The result of factorization is ()
A． B． C． D．
3. The result of factoring a polynomial factor is ()
A． B．
C． D．
4. The result of factoring is ()
A． B． C． D．
5. Factoring a polynomial into ()
A． B．
C． D．
6. In the range of rational number, factorization, the number of factors in the result is ()
A. 3 b.4 C.5 d.6
7. It is known that the area of a rectangle is, if one side is long, then the other side is long___________ .
8. Given that X and y are opposite to each other and = 4, then x = 1________ ,y＝________ .
9. Factorization: =________________ .
10. Calculation by factorization: =_____________ .
11. If known, then x =________ ,y＝__________ .
12. If it is known, then the value of the algebraic expression is_______________ .
15.4 answers for session 2:
1. B (dialing: can use the characteristics of the formula of square difference. First, there are two terms on the left that can be expressed in the form of square. The signs in front of these two terms are positive and negative.)
2. D (dial: original =)
3. D (dial: and then use the square difference formula)
4. D (dial: if there is a common factor, extract the common factor first, and then use the square difference formula)
5. D (dial: factorize the first two terms with the square difference formula, and then extract the common factor)
6. C (dial: =)
7、
8、 －
9、
10. - 12.996 (dial: original = =)
11、
12、8
Follow up training:
1、（ ）2＋20xy＋25 ＝( )2.
2. If known, then =__________ .
3. If it is known, then x + y =________ .
4. If it is a complete square, then the value of the real number m is ()
A. - 5 B.3 c.7 d.7 or - 1
5. If a binomial plus a monomial becomes a complete square, then such a monomial has ()
A. 1 B.2 C.3 D.4
6. Calculation by factorization: =_______________ .
7. Factorization in real numbers: =_____________________ .
8. Factorize the following
（1） （2）
（3） （4）
9. Factorization: = (), () - 20 (x + y) = ()
10. The result of factorization is_________________________ .
11. Given x + y = 7, xy = 10
(1) (2)
12. If, find the value of
15.4 answers for session 3:
1、2x 2x＋5y
2、
3、－2
4. D (tick: the middle term should be twice the product of X and 2, so M-3 = ± 4)
5. C (tick: if the two known terms are the sum of squares, the missing term should be twice the product ± 4x; if it is twice the product, the missing term is a square term; if 4 is twice the product, the missing term is, and the last one is a fraction, which does not meet the requirements)
6、90000
7、
8、（1） ；（2） ；（3） ；（4） 9、x＋y＋4 25 2x＋2y－5
10、
11、（1）∵x＋y＝7,xy＝10,∴ ,
∴ ,∴ ,∴ ＝58
（2）∵ ,∴ ,∴ ＝841
∴ ＝641
∴ ＝ ＝441
12、∵ ,∴ ,
∴ ＝ ＝－3×5＋7＝－8
1、 30 points for each choice
1. The following deformation from left to right is ()
A． B．
C． D．
2. Cannot be divided by which of the following numbers ()
A．2003 B．2002 C．2001 D．1001
3. Given M-N = 3, Mn = 1, then the value of is ()
A．5 B．7 C．9 D．11
4. Factoring a polynomial into ()
A． B．
C． D．
5. If 4x-3 is a factor of a polynomial, then a is equal to ()
A．－6 B．6 C．－9 D．9
2、 If you fill in carefully, you will feel relaxed (6 points for each question, 30 points in total)
6. Factorization: =______________________ .
7. The common factor of a polynomial is__________________ .
8. Using factorization method to calculate =__________________ .
9. If a polynomial is added with a monomial to make it the complete square of an integer, then the added monomial can be_______________________ (fill in a
RELATED INFORMATIONS
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- 4. How to factorize a cubic equation of one variable? Let me give you a cubic equation of one variable. How to factorize it? It's better to be easy to read and understand, rather than paste it casually.
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- 8. A mathematical factorization problem X ^ 8 + x ^ 6 + x ^ 4 + x ^ 2 + X + 1. Remember, it's factorization!
- 9. How to solve the problem of first 3 quadratic function and 1-variable quadratic equation? Find the coordinates of the image and X axis of the following quadratic function? Y = 1 / 2x * - x + 3
- 10. Elementary two problems of quadratic equation with one variable It is known that ax ② + BX + C = 0 has two roots, and a + C = 0 can find the relationship between the two roots
- 11. Use factorization to solve the following equation! ①﹙2x+3﹚²-2x-3=0 ②﹙2x-1﹚²-x²=0 ③2y﹙2+y﹚=-﹙y+2﹚ ④﹙y-1﹚²+2y﹙y-1﹚=0
- 12. 180°-x°=0.5x°
- 13. Quadratic equation of one variable 1. It is known that equation 2-root 5 is one root of equation xsquare-4x + C = 0, then what is the other root? 2. Solve the equation. Add 1 to each root of equation xsquare-4x-3 = 0; Then the new equation is? 3. If the equation AX ^ 2 + X-1 = 0 about X has real roots, then the range of a is 4. It is known that x = 1 is a solution of the quadratic equation AX ^ 2 + bx-40 = 0 of one variable, and a is not equal to B. find the value of a ^ 2-B ^ 2 divided by 2a-2b. 5. It is known that the equation x ^ 2-2 (M + 1) XM ^ 2 = 0 about X (1) When m takes what value, the equation has no real root (2) take a suitable non-zero integer for M. make the equation have two real roots, and find the square of their differences. 6. Given that x1.x2 is the relationship between two utilization coefficients and roots of 2x ^ 2 + 4x-3 = 0, find the value of x2 / X1 + X1 / x2
- 14. Given X & # 178; - X-1 = 0, what is the value of - X & # 179; + 2x & # 178; + 2013?
- 15. Several factorizations (square difference formula) 1) X square y square Z square 2) 9 / 4m square - 0101n square 3) 16 (a-b) square - 9 (a + b) square 4) 25p square - 49q square 5) 4A square - (B + C) square
- 16. The relationship between the root and coefficient of quadratic equation with one variable We know the equation x ^ 2 - (K + 1) x + 1 / 4K ^ 2 + 1 = 0 about X, if two real roots x, y of the equation satisfy |X | = y, find the value of K
- 17. Calculate 3x · (X & # 178; + X + 2) -a²·（a+b）+b·（a²-b²）
- 18. Factorization of group decomposition method 1.(a+2)(a+3)+(a+1)(a+4)+(a+1)(a+3)+(a+2)(a+4) 2.36x3-12x2-x + 1 / 3
- 19. How can a quadratic equation of two variables be transformed into a quadratic equation of one variable? Let's put it in common
- 20. What is the positional relationship between the line 3x 4y-13 = 0 and the circle (X-2) &# 178; (Y-3) &# 178; = 1?