Factor
1. 8a^2-2ab-21b^2
2. 6m^2-90m+324
3.5x(2-x)+4x(2x-5)-(3x-4)
4. 9(y-1)^2-25
5. 3x^2-27(2-x)^2
4. 9(y-1)^2 - 25 = [3(y-1)]^2 - 5^2
= [3(y-1)+5]*[3(y-1)-5]
= (3y-2)(3y-8)
5. 3x^2 - 27(2-x)^2
= 3[x^2 - 9(2-x)^2]
= 3[x^2 - [3(2-x)]^2]
= 3[(x+3(2-x)][x-3(2-x)]
= 3(x + 6 - 3x)(x - 6 + 3x)
= 3(-2x+6)(4x-6)
= 12(3-x)(2x-3)
It's late -- someone else do 1-3
To factor the given expressions, we can use various techniques depending on the type of expression. I will explain each step of the factoring process for each of the given expressions.
1. 8a^2 - 2ab - 21b^2:
To factor this quadratic expression, we can use the factoring method known as "Grouping." The steps are as follows:
- Find two numbers that multiply to give the product of the coefficient of the squared term (8) and the constant term (-21). In this case, the numbers are -4 and 5 because (-4)*(5) = -20, which is equal to the coefficient of the middle term (-2ab).
- Split the middle term (-2ab) using the two numbers we found in the previous step. We rewrite the middle term as (-4ab + 5ab).
- Factor by grouping: Group the terms with common factors together. In this case, the first two terms have the common factor 2a, and the last two terms have the common factor -7b.
- Factor out the common factors from each group separately. The expression becomes: 2a(4a - b) - 7b(4a - b).
- Notice that we have created a common factor of (4a - b). Factor out the common factor: (4a - b)(2a - 7b).
Thus, the factored form of the expression is (4a - b)(2a - 7b).
2. 6m^2 - 90m + 324:
To factor this quadratic expression, we can use the factoring method known as "Perfect Square Trinomial." The steps are as follows:
- Check if the expression has the form of a perfect square trinomial. In this case, it does because the first term (6m^2) is the square of the square root of the term, and the last term (324) is the square of the square root of the term.
- Rewrite the expression as the square of a binomial. In this case, the square root of the first term (6m^2) is (sqrt(6m))^2 = 6m, and the square root of the last term (324) is (sqrt(324))^2 = 18.
- Write the expression as a perfect square trinomial: (6m - 18)^2.
Thus, the factored form of the expression is (6m - 18)^2.
3. 5x(2 - x) + 4x(2x - 5) - (3x - 4):
To factor this expression, we can simplify it first and then combine like terms. The steps are as follows:
- Distribute the multiplication across each term: 10x - 5x^2 + 8x^2 - 20x - 3x + 4.
- Combine like terms: -5x^2 + 8x^2 + 10x - 20x - 3x + 4.
- Simplify the expression: 3x^2 - 13x + 4.
Now, we can factor this quadratic expression using the factoring method known as "Trial and Error" or "Grouping."
- Find two numbers that multiply to give the product of the coefficient of the squared term (3) and the constant term (4). In this case, the numbers are 1 and 4 because (1)*(4) = 4, which is equal to the coefficient of the constant term (4).
- Split the middle term (-13x) using the two numbers we found in the previous step. We rewrite the middle term as (-x - 12x).
- Factor by grouping: Group the terms with common factors together. In this case, the first two terms have the common factor x, and the last two terms have the common factor -4.
- Factor out the common factors from each group separately. The expression becomes: x(3x - 1) - 4(3x - 1).
- Notice that we have created a common factor of (3x - 1). Factor out the common factor: (3x - 1)(x - 4).
Thus, the factored form of the expression is (3x - 1)(x - 4).
4. 9(y - 1)^2 - 25:
To factor this expression, we can use the factoring method known as "Difference of Squares." The steps are as follows:
- Recognize that the expression has two perfect squares being subtracted: (y - 1)^2 and 5^2.
- Use the formula for the difference of squares: a^2 - b^2 = (a + b)(a - b).
- Apply the formula, replacing a with (y - 1) and b with 5: (y - 1 + 5)(y - 1 - 5).
- Simplify the expressions: (y + 4)(y - 6).
Thus, the factored form of the expression is (y + 4)(y - 6).
5. 3x^2 - 27(2 - x)^2:
To factor this expression, we can start by simplifying it first. The steps are as follows:
- Expand the squared term using the formula (a - b)^2 = a^2 - 2ab + b^2. In this case, (2 - x)^2 = (2)^2 - 2(2)(-x) + (-x)^2 = 4 + 4x + x^2.
- Rewrite the expression using the simplified form of the squared term: 3x^2 - 27(4 + 4x + x^2).
- Distribute the multiplication across each term: 3x^2 - 108 - 108x - 27x^2.
- Combine like terms: -24x^2 - 108x - 108.
Now, we can factor this quadratic expression using the factoring method known as "Trial and Error" or "Grouping."
- Find a common factor for all the terms, which is -12: -12(2x^2 + 9x + 9).
- Factor out the common factor: -12(2x^2 + 9x + 9).
- Notice that the quadratic trinomial 2x^2 + 9x + 9 cannot be factored further using integer factors.
Thus, the factored form of the expression is -12(2x^2 + 9x + 9).