Question

1 вариант1). Преобразуйте в многочлен:а). ( а – 3 )2 ; б). ( 2х + у )2 ; в). ( 5в – 4х )( 5в + 4х ). 2). Упростите выражение:( а – 9)2 – ( 81 + 2а)3). Разложите на множители:а). х 2 – 25 ; б). ав 2 – ас 2 ;в). – 3а 2 – 6ав – 3ав 2 .4). Решите уравнение:( 2 – х )2 – х( х + 1,5 ) = 45). Выполните действия:а). (у2 – 2а)( 2а + у2); б). ( 3х2 + х)2;в). ( 2 + т)2( 2 – т)2 6). Разложите на множители:а). 4х2у2 – 9а4; б). 25а 2 – ( а + 3 )2 ; б). 27 а 3 + в 3

Answer 1

1). Преобразуйте в многочлен:
$a)~ (a-3)^2=a^2-2\cdot a\cdot3+3^2=a^2-6a+9\\ \\ b)~ (2x+y)^2=(2x)^2+2\cdot2x\cdot y+y^2=4x^2+4xy+y^2\\ \\ c)~ (5b-4x)(5b+4x)=(5b)^2-(4x)^2=25b^2-16x^2$

2). Упростите выражение:
$(a-9)^2-(81+2a)=a^2-18a+81-81-2a=a^2-20a$

3). Разложите на множители:
a) Применим здесь формулу разности квадратов:
   $x^2-25=x^2-5^2=(x-5)(x+5)$

$b)~ ab^2-ac^2=a(b^2-c^2)=a(b-c)(b+c)\\ \\ c)~ -3a^2-6ab-3ab^2=-3a(a+2b+b^2)$

4). Решите уравнение:
$(2-x)^2-x(x+1.5)=4\\ \\ 4-4x+x^2-x^2-1.5x=4\\ \\ -5.5x=0\\ \\ x=0$

5). Выполните действия:

$a)~ (y^2-2a)(2a+y^2)=y^4-4a^2\\ \\ b)~ (3x^2+x)^2=9x^4+6x^3+x^2\\ \\ c)~ (2+t)^2(2-t)^2=(4-t^2)^2=16-8t^2+t^4$

6). Разложите на множители:
$a)~ 4x^2y^2-9a^4=(2xy)^2-(3a^2)^2=(2xy+3a^2)(2xy-3a^2)\\ \\ b)~ 25a^2-(a+3)^2=(5a-a-3)(5a+a+3)=3(4a-3)(2a+1)\\ \\ c)~27a^3+b^3=(3a+b)(9a^2-3ab+b^2)$

Answer 2

1) Преобразование в многочлен

а) *(а – 3)^2* To expand the expression *(a – 3)^2*, we can use the formula for squaring a binomial: *(a – b)^2 = a^2 – 2ab + b^2*.

Expanding *(a – 3)^2*: *(a – 3)^2 = a^2 - 2(a)(3) + 3^2* Simplifying: *(a – 3)^2 = a^2 - 6a + 9*

б) *(2х + у)^2* The expansion of *(2x + y)^2* can be done using the same formula.

Expanding *(2x + y)^2*: *(2x + y)^2 = (2x)^2 + 2(2x)(y) + (y)^2* Simplifying: *(2x + y)^2 = 4x^2 + 4xy + y^2*

в) *(5в – 4х)(5в + 4х)* To expand the expression *(5v - 4x)(5v + 4x)*, we can use the formula for multiplying two binomials: *(a - b)(a + b) = a^2 - b^2*.

Expanding *(5v - 4x)(5v + 4x)*: *(5v - 4x)(5v + 4x) = (5v)^2 - (4x)^2* Simplifying: *(5v - 4x)(5v + 4x) = 25v^2 - 16x^2*

2) Упрощение выражения

*(а – 9)^2 – (81 + 2а)^3* To simplify the expression *(a - 9)^2 - (81 + 2a)^3*, we need to expand and simplify each term.

Expanding *(a - 9)^2*: *(a - 9)^2 = a^2 - 2(a)(9) + 9^2* Simplifying: *(a - 9)^2 = a^2 - 18a + 81*

Expanding *(81 + 2a)^3*: *(81 + 2a)^3 = (81)^3 + 3(81)^2(2a) + 3(81)(2a)^2 + (2a)^3* Simplifying: *(81 + 2a)^3 = 531441 + 4374a + 108a^2 + 8a^3*

Substituting the expanded expressions back into the original expression: *(a - 9)^2 - (81 + 2a)^3 = (a^2 - 18a + 81) - (531441 + 4374a + 108a^2 + 8a^3)* Simplifying: *(a - 9)^2 - (81 + 2a)^3 = -8a^3 - 108a^2 - 4554a - 531360*

3) Разложение на множители

а) *х^2 - 25* To factor the expression *x^2 - 25*, we can use the difference of squares formula: *(a^2 - b^2) = (a + b)(a - b)*.

Factoring *x^2 - 25*: *x^2 - 25 = (x + 5)(x - 5)*

б) *ав^2 - ас^2* To factor the expression *av^2 - ac^2*, we can factor out the common factor *a* and then apply the difference of squares formula.

Factoring *av^2 - ac^2*: *av^2 - ac^2 = a(v^2 - c^2)* Applying the difference of squares formula: *av^2 - ac^2 = a(v + c)(v - c)*

в) *-3a^2 - 6av - 3av^2* To factor the expression *-3a^2 - 6av - 3av^2*, we can factor out the common factor *-3a*.

Factoring *-3a^2 - 6av - 3av^2*: *-3a^2 - 6av - 3av^2 = -3a(a + 2v + v^2)*

4) Решение уравнения

*(2 - x)^2 - x(x + 1.5) = 4* To solve the equation *(2 - x)^2 - x(x + 1.5) = 4*, we can simplify and then solve for *x*.

Simplifying: *(2 - x)^2 - x(x + 1.5) = 4* Expanding *(2 - x)^2*: *(2 - x)^2 = 4 - 4x + x^2* Expanding *x(x + 1.5)*: *x(x + 1.5) = x^2 + 1.5x* Substituting the simplified expressions back into the original equation: *(4 - 4x + x^2) - (x^2 + 1.5x) = 4* Simplifying: *4 - 4x + x^2 - x^2 - 1.5x = 4* Combining like terms: *-5.5x = 0* Solving for *x*: *x = 0*

5) Выполнение действий

а) *(у^2 - 2а)(2а + у^2)* To perform the operation *(у^2 - 2а)(2а + у^2)*, we can use the distributive property to multiply each term.

Expanding *(у^2 - 2а)(2а + у^2)*: *(у^2 - 2а)(2а + у^2) = у^2(2а + у^2) - 2а(2а + у^2)* Simplifying: *(у^2 - 2а)(2а + у^2) = 2ау^2 + у^4 - 4а^2 - 2ау^2* Combining like terms: *(у^2 - 2а)(2а + у^2) = у^4 - 4а^2*

б) *(3х^2 + х)^2* To perform the operation *(3х^2 + х)^2*, we can use the distributive property to square each term.

Expanding *(3х^2 + х)^2*: *(3х^2 + х)^2 = (3х^2)^2 + 2(3х^2)(х) + (х)^2* Simplifying: *(3х^2 + х)^2 = 9х^4 + 6х^3 + х^2*

в) *(2 + т)^2(2 - т)^2* To perform the operation *(2 + т)^2(2 - т)^2*, we can use the difference of squares formula.

Expanding *(2 + т)^2(2 - т)^2*: *(2 + т)^2(2 - т)^2 = (2^2 - т^2)^2* Simplifying: *(2 + т)^2(2 - т)^2 = (4 - т^2)^2* Expanding *(4 - т^2)^2* using the difference of squares formula again: *(4 - т^2)^2 = (4 + т)(4 - т)* Simplifying: *(2 + т)^2(2 - т)^2 = (16 - т^2)*