1.
X^{2 }+ x – 90
Correct Answer
B. (x – 9)(x + 10)
Explanation
The given expression is a quadratic expression in the form of x^2 + x - 90. To factorize this expression, we need to find two numbers whose sum is equal to the coefficient of x (1) and whose product is equal to the constant term (-90). The numbers that satisfy these conditions are -9 and 10. Therefore, the correct factorization of the expression is (x - 9)(x + 10).
2.
X^{2 }+ x – 78
Correct Answer
D. None of these
3.
X^{2} – 6x + 8
Correct Answer
C. (x – 4)(x – 2)
Explanation
The given expression is a quadratic trinomial in the form of x^2 - 6x + 8. To factorize this trinomial, we need to find two binomials whose product equals the given trinomial. By using the FOIL method, we can determine that the binomials (x - 4) and (x - 2) multiply to give x^2 - 6x + 8. Therefore, the correct answer is (x - 4)(x - 2).
4.
5x^{2} + 10x + 20
Correct Answer
C. 5(x^2 + 2x + 4)
Explanation
The given expression can be factored as 5(x^2 + 2x + 4). This is the correct answer because it is the only option that correctly represents the factored form of the expression. The expression cannot be factored further, so the other options are incorrect.
5.
X^{2 }+ x – 90
Correct Answer
B. (x – 9)(x + 10)
Explanation
The given expression can be factored into (x - 9)(x + 10). This can be determined by finding two numbers that multiply to -90 and add up to 1 (the coefficient of x). The numbers that satisfy this condition are -9 and 10, which gives us the factored form (x - 9)(x + 10).
6.
X^{2} – 6x + 8
Correct Answer
C. (x – 4)(x – 2)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factorize it, we need to find two binomials in the form of (x + m)(x + n) where m and n are the factors of the constant term c (in this case, 8) that add up to the coefficient of the middle term b (in this case, -6). The factors of 8 are 1, 2, 4, and 8. The only combination that satisfies the condition is (x - 4)(x - 2), which is the correct answer.
7.
X^{2 }+ x – 870
Correct Answer
D. None of these
Explanation
The given expression, x2 + x - 870, cannot be factored into the given options (x - 4)(x - 2) or (x + 4)(x + 2). Therefore, the correct answer is "None of these".
8.
5x^{2} + 10x + 20
Correct Answer
C. 5(x^2 + 2x + 4)
Explanation
The given expression can be factored as 5(x^2 + 2x + 4). This is because the expression follows the pattern of a quadratic trinomial, where the coefficient of x^2 is 1, the coefficient of x is 2, and the constant term is 4. The factored form represents the expression in terms of its factors, which in this case is 5 multiplied by the quadratic trinomial (x^2 + 2x + 4).
9.
2x^{2} + 2x – 4
Correct Answer
A. 2(x – 1)(x + 2)
Explanation
The given expression can be factored as 2(x - 1)(x + 2). This can be determined by using the distributive property of multiplication over addition/subtraction. By multiplying 2 with each term inside the parentheses, we get 2x - 2 and 4. Therefore, the correct answer is 2(x - 1)(x + 2).
10.
15x^{2} – 27x – 6
Correct Answer
B. 3(5x + 1)(x – 2)
Explanation
The given expression can be factored as follows: 15x^2 - 27x - 6 = 3(5x^2 - 9x - 2) To factorize the expression further, we can look for two numbers whose product is equal to the product of the coefficient of x^2 term and the constant term (-10) and whose sum is equal to the coefficient of the x term (-9). These numbers are -10 and 1. Therefore, we can rewrite the expression as: 3(5x^2 - 10x + x - 2) = 3(5x(x - 2) + 1(x - 2)) = 3(5x + 1)(x - 2) Hence, the correct answer is 3(5x+ 1)(x- 2).