Answer:
1. 2(x + 2)(x - 1).
2. (3x + 4)(2x - 5).
3. (2x + 1)(x - 3).
Explanation:
To factor 2x^2 + 2x - 4, we can start by factoring out the greatest common factor, which is 2:
2x^2 + 2x - 4 = 2(x^2 + x - 2)
Next, we need to factor the quadratic expression inside the parentheses, which can be done using the product-sum method or by recognizing that the expression is a quadratic trinomial of the form x^2 + bx + c. Since the coefficient of x^2 is 1, we can easily see that the two factors must have the form (x + ?)(x + ?), where the question marks represent the two missing numbers we need to find.
To find these numbers, we can look for two numbers whose product is -2 (the constant term) and whose sum is 1 (the coefficient of x). These numbers are 1 and -2:
x^2 + x - 2 = (x + 2)(x - 1)
Therefore, we can write:
2x^2 + 2x - 4 = 2(x^2 + x - 2) = 2(x + 2)(x - 1)
So the factored form of the polynomial is 2(x + 2)(x - 1).
To factor 6x^2 - 7x - 20, we can again use the product-sum method or the quadratic formula. For simplicity, we'll use the product-sum method here. We need to look for two numbers whose product is -120 (the product of the leading coefficient 6 and the constant term -20) and whose sum is -7 (the coefficient of x). These numbers are -15 and 8:
6x^2 - 7x - 20 = 6x^2 - 15x + 8x - 20
= 3x(2x - 5) + 4(2x - 5)
= (3x + 4)(2x - 5)
Therefore, the factored form of the polynomial is (3x + 4)(2x - 5).
To factor 2x^2 - 5x - 3, we can again use the product-sum method or the quadratic formula. We need to look for two numbers whose product is -6 (the product of the leading coefficient 2 and the constant term -3) and whose sum is -5 (the coefficient of x). These numbers are -2 and 3:
2x^2 - 5x - 3 = 2x^2 - 6x + x - 3
= 2x(x - 3) + 1(x - 3)
= (2x + 1)(x - 3)
Therefore, the factored form of the polynomial is (2x + 1)(x - 3).