1.
3a^2+7a+2
Correct Answer
A. (a+2)(3a+1)
Explanation
The given expression is a quadratic trinomial. To factorize it, we need to find two binomials whose product is equal to the given expression. By analyzing the coefficients, we can determine that the factors are likely to be of the form (a + x)(3a + y). We can then expand this expression to obtain 3a^2 + 7a + 2. By comparing the coefficients of the expanded expression with the given expression, we find that x = 2 and y = 1. Therefore, the correct answer is (a + 2)(3a + 1).
2.
Factor 6x^2 + 7x + 2
Correct Answer
B. (2x + 1)(3x + 2)
Explanation
The given expression can be factored as (2x + 1)(3x + 2). This can be determined by using the FOIL method or by finding the factors of the constant term (2) and the leading coefficient (6). The factors of 2 are 1 and 2, and the factors of 6 are 1, 2, 3, and 6. By trying different combinations, we can find that (2x + 1)(3x + 2) is the correct factorization.
3.
Factor 8x^2 + 8x - 6
Correct Answer
C. 2(2x + 3)(2x -1)
Explanation
The given expression is a quadratic trinomial. To factor it, we can use the factoring method. By looking at the coefficients, we can see that the greatest common factor is 2. Factoring out 2, we are left with 4x^2 + 4x - 3. Now, we need to find two binomials that multiply to give us 4x^2 - 3. The binomials that satisfy this are (2x + 3) and (2x - 1). Therefore, the factored form of the expression is 2(2x + 3)(2x - 1).
4.
Factor 12x^2 - 17x - 5
Correct Answer
A. (4x + 1)(3x - 5)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factor it, we need to find two binomials that when multiplied together will result in the given trinomial. In this case, the factors are (4x + 1) and (3x - 5). When these two binomials are multiplied together using the distributive property, we get 12x^2 - 17x - 5, which matches the original expression. Therefore, the correct answer is (4x + 1)(3x - 5).
5.
Factor 20x^2 - 46x + 24
Correct Answer
D. 2(5x - 4)(2x - 3)
Explanation
The given expression can be factored using the method of factoring by grouping. First, we can factor out the greatest common factor of the terms, which is 2. This gives us 2(10x^2 - 23x + 12). Next, we look for two numbers that multiply to give 12 and add up to -23. These numbers are -4 and -3. So, we can rewrite the expression as 2(10x^2 - 4x - 3x + 12). Now, we can group the terms and factor out the common factors from each group. This gives us 2[(10x^2 - 4x) + (-3x + 12)]. Factoring out x from the first group and -3 from the second group gives us 2[2x(5x - 2) - 3(5x - 2)]. Finally, we can factor out the common factor of (5x - 2) from both terms, which gives us the final answer of 2(5x - 4)(2x - 3).
6.
Factor 6x^2 + 15x + 9
Correct Answer
B. 3(2x + 3)(x + 1)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factorize it, we can use the factoring method. In this case, we can factor out the greatest common factor of 3. This gives us 3(2x^2 + 5x + 3). Now we need to find two binomials that multiply together to give us 2x^2 + 5x + 3. The binomials that satisfy this condition are (2x + 3) and (x + 1). Therefore, the correct factorization is 3(2x + 3)(x + 1).
7.
Factor 18t^2 - 24t + 6
Correct Answer
A. 6(3t - 1)(t - 1)
Explanation
The given expression is a quadratic trinomial. To factor it, we can look for two binomials that multiply together to give us the trinomial. In this case, we can factor out a common factor of 6 from all the terms to simplify the expression. Then, we can use the distributive property to multiply the binomials. The factors of the trinomial are (3t - 1) and (t - 1), so the correct answer is 6(3t - 1)(t - 1).
8.
Factor 6x^2 + 23x + 7
Correct Answer
D. (3x + 1)(2x + 7)
Explanation
The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factor it, we need to find two binomials that multiply together to give us the original expression. In this case, we can use the FOIL method to expand (3x + 1)(2x + 7) and see if it matches the original expression. When we multiply the first terms (3x)(2x), we get 6x^2. When we multiply the outer terms (3x)(7), we get 21x. When we multiply the inner terms (1)(2x), we get 2x. Finally, when we multiply the last terms (1)(7), we get 7. Combining these terms, we get 6x^2 + 23x + 7, which matches the original expression. Therefore, the answer is (3x + 1)(2x + 7).
9.
Factor 3x^2 + 5x - 2
Correct Answer
C. (3x - 1)(x + 2)
Explanation
The given expression is a quadratic trinomial. To factor it, we need to find two binomials whose product is equal to the expression. The correct answer is (3x - 1)(x + 2) because when we multiply these two binomials, we get 3x^2 + 5x - 2, which matches the original expression.
10.
Factor 18x^2 + 3x - 10
Correct Answer
C. 6x + 5)(3x - 2)