Question

1. Преобразуйте в многочлен стандартного вида: а) (5a − b)(5a + b); б) (х-7)^2-3(х+5) ( 5) . 2. Разложите на множители: а) 4x + 4y − (xz + y); б) m^3− m^2−9 m +9; в) 4t^3−(t − 5)^2 3

Answer 1

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Answer 2

1. Преобразуйте в многочлен стандартного вида:

а) (5a - b)(5a + b)

To expand this expression, we can use the formula for multiplying two binomials:

(a - b)(a + b) = a^2 - ab + ab - b^2 = a^2 - b^2

Using this formula, we can expand the given expression:

(5a - b)(5a + b) = (5a)^2 - (b)^2 = 25a^2 - b^2

Therefore, the expanded form of the expression (5a - b)(5a + b) is 25a^2 - b^2.

б) (x - 7)^2 - 3(x + 5) + 5

To expand this expression, we first need to apply the distributive property to the term -3(x + 5):

-3(x + 5) = -3 * x - 3 * 5 = -3x - 15

Next, we can expand the squared term (x - 7)^2 using the formula (a - b)^2 = a^2 - 2ab + b^2:

(x - 7)^2 = x^2 - 2 * x * 7 + 7^2 = x^2 - 14x + 49

Finally, we can substitute the expanded expressions back into the original expression:

(x - 7)^2 - 3(x + 5) + 5 = x^2 - 14x + 49 - 3x - 15 + 5

Combining like terms, we get:

x^2 - 14x + 49 - 3x - 15 + 5 = x^2 - 17x + 39

Therefore, the expanded form of the expression (x - 7)^2 - 3(x + 5) + 5 is x^2 - 17x + 39.

2. Разложите на множители:

а) 4x + 4y - (xz + y)

To factor this expression, we can factor out the common terms in each group:

4x + 4y - (xz + y) = 4(x + y) - (x - y)z

Now, we can factor out the common factor in the first group, which is 4:

4(x + y) - (x - y)z = 4(x + y) - z(x - y)

Finally, we can factor out the common factor in the second group, which is (x - y):

4(x + y) - z(x - y) = (x - y)(4 + z)

Therefore, the factored form of the expression 4x + 4y - (xz + y) is (x - y)(4 + z).

б) m^3 - m^2 - 9m + 9

To factor this expression, we can look for common factors in each group:

m^3 - m^2 - 9m + 9 = m^2(m - 1) - 9(m - 1)

Now, we can factor out the common factor in each group, which is (m - 1):

m^2(m - 1) - 9(m - 1) = (m - 1)(m^2 - 9)

Next, we can further factor the difference of squares, m^2 - 9:

(m - 1)(m^2 - 9) = (m - 1)(m - 3)(m + 3)

Therefore, the factored form of the expression m^3 - m^2 - 9m + 9 is (m - 1)(m - 3)(m + 3).

в) 4t^3 - (t + 5)^2 + 3

To factor this expression, we can first expand the squared term (t + 5)^2:

(t + 5)^2 = t^2 + 2 * t * 5 + 5^2 = t^2 + 10t + 25

Now, we can substitute the expanded expression back into the original expression:

4t^3 - (t + 5)^2 + 3 = 4t^3 - (t^