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Factorization: x 3 + 6 x 2 + 11 x + 6
x 3 + 6 x 2 + 11 x + 6 =（x^3+x²）+（5x²+5x）+（6x+6）
=x²(x+1)+5x(x+1)+6(x+1)
=(x+1)(x²+5x+6)
=(x+1)(x+2)(x+3)
x^3 + 6 x^2 + 11x + 6
= x (x+2)^2 +(x+2)(x+3)
= (x+2)(x^2+2x+2x+3)
= (x+2)(x+1)(x+3)
You can use factor division to try, first divide by X + 1, see if it can be divided completely, and then see if the rest can be decomposed.
x^2 +5x + 6
/--------------------... unfold
x^3 + 6 x^2 + 11x + 6
= x (x+2)^2 +(x+2)(x+3)
= (x+2)(x^2+2x+2x+3)
= (x+2)(x+1)(x+3)
You can use factor division to try, first divide by X + 1, see if it can be divided completely, and then see if the rest can be decomposed.
x^2 +5x + 6
/------------------------
x+1 / x^3 + 6 x^2 + 11x + 6
x^3 + x^2
------------------------------------------
5x^2 + 11 x + 6
5x^2 + 5x
------------------------------
6x + 6
6x + 6
------------------------------Put it away
(x^2)-(x^4)+12
Take x ^ 2 as a whole:
Original = - [(x ^ 2) ^ 2-x ^ 2-12]
=-(X^2-4)(X^2+3)
=-(X+2)(X-2)(X^2+3).
-The factorization of 3xy & sup2; - 8yx - 2x is the simplest
Analysis:
Decomposition in the range of rational numbers
-3xy²-8yx-2x=-x(3y^2+8y+2)
Decomposing in real numbers
-3xy²-8yx-2x=-x(3y^2+8y+2)
=-3x[(y-4/3)^2-10/9]
=-3x(y-4/3+√10/3)(y-4/3-√10/3)
-3xy²-8yx-2x
=-x(3y²+8y+2)
Let x ^ 4 + x ^ 2 = y, then the original formula = (y-4) (y-4) (y-4) (y + 3) (y-4) (y + 3) + 10 = y ^ 2-2-2-2-y-2 = (y-4) (y-4) (y-4) (y-4) (y-4) (y-4) (y + 3 + 3) + (x ^ 4 + x ^ 4 + x ^ 2-2-4) (x ^ 4 + x ^ 2 + 3 + 3) + 10. Let x ^ 4 + x ^ 4 + x ^ 2 = x ^ 2 = y, then the original formula = (y-4) (y-4) (y-4) (Y (y-4) (Y (y-4) (x ^ 4) (x ^ 4 + x ^ 2) (x ^ 4 + x ^ 4 + x ^ 2) (x ^ 4 + x ^ 4 + x ^ 2) (x ^ 4 + x ^ 4 + x ^ 4 + x ^ 2 + x ^ 2 (x ^ 2) (x ^ 2) (x ^ 2 (x ^ 2 strength,
(X^4+X^2-2)=（X^2+2）（x^2-1）=（X^2+2）(x+1)(x-1)
The first step is to look at the factor of x ^ 2
x^4+x^2-2=(x^2+2)(x^2-1)=(x^2+2)(x+1)(x-1)
Factorization, cross formula
(x ^ 4 + x ^ 2-2) (x ^ 4 + x ^ 2 + 1) = (x ^ 2 + 2) (x + 1) (x-1) (x ^ 4 + x ^ 2 + 1)
This step decomposes (x ^ 4 + x ^ 2-2)
Where x ^ 2 is taken as a whole
(X^4+X^2-2)
=（X^2+2）（X^2-1）
=（X^2+2）（x-1)（x+1)
So (x ^ 4 + x ^ 2-2) (x ^ 4 + x ^ 2 + 1)
=(X^2+2)(X+1)(X-1)(X^4+X^2+1)
(X^4+X^2-2)(X^4+X^2+1)=(X^2+2)(X+1)(X-1)(X^4+X^2+1)
This step: the main reason is that the front has changed, that is, (x ^ 4 + x ^ 2-2) = (x ^ 2 + 2) (x + 1) (x-1)
This process is like this. Let's set X ^ 2 = y, then (x ^ 4 + x ^ 2-2) = y ^ 2 + Y-2 = (y + 2) (Y-1)
Replace y = x ^ 2. Have (x ^ 2 + 2) (x ^ 2-1) = (x ^ 2 + 2... Expand
(X^4+X^2-2)(X^4+X^2+1)=(X^2+2)(X+1)(X-1)(X^4+X^2+1)
This step: the main reason is that the front has changed, that is, (x ^ 4 + x ^ 2-2) = (x ^ 2 + 2) (x + 1) (x-1)
This process is like this. Let's set X ^ 2 = y, then (x ^ 4 + x ^ 2-2) = y ^ 2 + Y-2 = (y + 2) (Y-1)
Replace y = x ^ 2. There are (x ^ 2 + 2) (x ^ 2-1) = (x ^ 2 + 2) (x + 1) (x-1)
So (x ^ 4 + x ^ 2-2) (x ^ 4 + x ^ 2 + 1) = (x ^ 2 + 2) (x + 1) (x-1) (x ^ 4 + x ^ 2 + 1) close
Quadratic equations of one variable with roots 2 and - 5
Square of X + 3x-10 = 0
X2+3X-10=0
xx+3x-10
Eighth grade volume 1 factorization exercises (PEP)
1、 Multiple choice questions
1. Given that Y2 + my + 16 is a complete square, then the value of M is ()
A．8 B．4 C．±8 D．±4
2. The following polynomials can be factorized by the complete square formula ()
A．x2-6x-9 B．a2-16a+32 C．x2-2xy+4y2 D．4a2-4a+1
3. The following formulas belong to the correct factorization ()
A．1+4x2=（1+2x）2 B．6a-9-a2=-（a-3）2
C．1+4m-4m2=（1-2m）2 D．x2+xy+y2=（x+y）2
4. Factoring x4-2x2y2 + Y4, the result is ()
A．（x-y）4 B．（x2-y2）4 C．[（x+y）（x-y）]2 D．（x+y）2（x-y）2
2、 Fill in the blanks
5. Given that 9x2-6xy + k is a complete square, then the value of K is________ .
6．9a2+（________） +25b2=（3a-5b）2
7．-4x2+4xy+（_______） =-（_______） .
8. Given A2 + 14a + 49 = 25, then the value of a is_________ .
3、 Answer questions
9. Factorize the following formulas:
①a2+10a+25 ②m2-12mn+36n2
③xy3-2x2y2+x3y ④（x2+4y2）2-16x2y2
10. Given x = - 19, y = 12, find the value of the algebraic formula 4x2 + 12xy + 9y2
11. Given that X-Y + 1 and x 2 + 8 x + 16 are opposite numbers, find the value of x 2 + 2XY + y 2
4、 Inquiry questions
12. Do you know the holistic thinking in mathematics? In solving problems, if you put your attention and focus on the whole of the problem, think, associate and explore in many directions, think and deform as a whole, and determine the problem-solving strategies from different aspects, the problem can be solved quickly
Can you factorize the following equations in a holistic way?
①（x+2y）2-2（x+2y）+1 ②（a+b）2-4（a+b-1）
Reference answer:
1. C 2. D 3. B 4. D 5. Y2 6. - 30ab 7. - Y2; 2x-y 8. - 2 or - 12
9．①（a+5）2；②（m-6n）2；③xy（x-y）2；④（x+2y）2（x-2y）2
10．4
11．49
12．①（x+2y-1）2；②（a+b-2）2
It is known that the parabola y = a (X-H) 2 has the maximum value when x = 2. This parabola passes through the point (1, - 3). The analytical formula of the parabola is obtained and it is pointed out that when x is the value, y increases with X
And decrease
When x = 2, y has a maximum,
X = 2 is its axis of symmetry, H = 2,
Substituting points (1, - 3) into the analytical expression, we get the following results:
y=a﹙x－2﹚²,
∴a﹙1－2﹚²=－3,
∴a=－3,
The analytic formula is: y = - 3 (X-2) &;,
When x > 2, y decreases with the increase of X
The relationship between the root and coefficient of bivariate equation
A more comprehensive answer
Ax ^ 2 + BX + C = 0 (a ≠ 0) two x1, X2 then
x1+x2=-b/a
x1x2=c/a
The two real roots of ax ^ 2 + BX + C = 0 are
x1=[-b+√(b^2-4ac)]/(2a), x2=[-b -√(b^2-4ac)]/(2a)
When B ^ 2-4ac < 0, the equation has no real roots
When B ^ 2-4ac = 0, the equation has two equal real roots
When B ^ 2-4ac > 0, the equation has two unequal real roots
Weida theorem: X1 + x2 = - B / A, X1 * x2 = C / A
Factoring factor (1) 3x ^ 2 + 5xy + y ^ 2 in the range of real numbers
（2）3x^2y^2+10xy+5
(1) In this paper, we are going to solve the quadratic equation 3x (x) for the second time of X, and we are going to solve the second equation 3x \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\the + 5 solution is about XY
1. Fill in the blanks: 2. (A-3) (3-2a) = (3-A) (3-2a); 12. If m2-3m + 2 = (M + a) (M + b), then a =, B =; 15. When m =, X2 + 2 (M-3) x + 25 is a complete square. 2. Multiple choice questions: 1 In the factorization result of 6284, the correct one is a.a2b + 7ab-b = B (A2 + 7a) b.3x2y-3xy-6y = 3Y (X... expansion)
1. Fill in the blanks: 2. (A-3) (3-2a) = (3-A) (3-2a); 12. If m2-3m + 2 = (M + a) (M + b), then a =, B =; 15. When m =, X2 + 2 (M-3) x + 25 is a complete square. 2. Multiple choice questions: 1 In the factorization result of 6284, the correct one is a. A2B + 7ab-b = B (A2 + 7a) B. 3x2y-3xy-6y = 3Y (X-2) (x + 1) C. 8xyz-6x2y2 = 2xyz (4-3xy) d. - 2A2 + 4ab-6ac = - 2A (a + 2b-3c) 2. The factorization factor of polynomial m (n-2) - M2 (2-N) is equal to A. (n-2) (M + m2) B. (n-2) (m-m2) C.M (n-2) (M + 1) D.M (n-2) (m-1) 3. In the following equations, a belongs to factorization: A.A (X-Y) + B (M + n) = ax + BM ay + BN b.a2-2ab + B2 + 1 = (a-b) 2 + 1 C. - 4a2 + 9b2 = (- 2A + 3b) (2a + 3b) d.x2-7x-8 = x (X-7) - 8 4 If 9x2 + MXY + 16y2 is a complete square, then the value of M is 〔 A. - 12b. ± 24C. 12D. ± 126. The polynomial an + 4-an + 1 is decomposed into 〔 A.an (a4-a) b.an-1 (a3-1) C.an + 1 (A-1) (a2-a + 1) d.an + 1 (A-1) (A2 + A + 1) 7. If A2 + a = - 1g, then the value of A4 + 2a3-3a2-4a + 3 is 〔 A.8 B.7 C.10 d.128. Given that x2 + Y2 + 2x-6y + 10 = 0, then the values of X and y are 〔 a.x = 1, y = 3 b.x = 1, y = - 3 c.x = - 1, y = 3 D.X = 1, y = 3 D.X = 1, y = 3 d (M + 1) 4 (M + 2) 2 b. (m-1) 2 (m-2) 2 (M2 + 3m-2) C. (M + 4) 2 (m-1) 2 (m-1) 2 d. (M + 1) 2 (M + 2) 2 (M2 + 3m-2) 210. The factorization of x2-7x-60 yields A. (X-10) (x + 6) B. (x + 5) (x-1) 2) We decompose the factor 3x (2) (2x) (2) (y) 3x (2) a (3) y (3) a (2) (4) y (3) a (2) (4) a (2) (4) y (3) a (2) (4) a (2) (4) a (2) (4) y (3) a (2) (4) a (2) (2) (4) y (3) (2) (2) (2) (2) (2) (4) a (2) (4) y (3) a (2) (2) (4) a (2) (4) y (2) (2) (3) a (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (3) y) a (3) a (3) a (2) (2) ( ）(a-3b) C. (a + 11b) (a-3b) d. (a-11b) (a + 3b) 13. The factorization of x4-3x2 + 2 gives 〔 A. (x2-2) (x2-1) B. (x2-2) (x + 1) (x-1) C. (x2 + 2) (x2 + 1) d. (x2 + 2) (x + 1) (x-1) 14. The factorization of polynomial x2-ax-bx + AB is 〕 a - (x + a) (x + b) B. (x-a) (x + b) C. (x-a) (X-B) d. (x + a) (x + b) 15. A quadratic trinomial of X, whose coefficient of x2 term is 1 and constant term is - 12, can be decomposed into factors. Such a quadratic trinomial is 〔 a.x2-11x-12 or x2 + 11x-12 b.x2-x-1 〕 Put it away
Let y = - X & # 178; + BX + C be 1
(1) The analytic formula of parabola
y=-x²+6x-5
(2) If the line y = KX + B (K ≠ 0) intersects the parabola at points a (3 / 2, m) and B (4, n), the analytical expression of the line can be obtained
m=7/4,n=3
Straight line y = 0.5x + 1
(3) Let the lines X = t and x = t + 2 parallel to the y-axis be called segments AB and E, f respectively, and the intersecting quadratic function and h, g respectively
① Find the value range of T
3/2
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