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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 whose product equals the given expression. In this case, the binomials are (x + 5) and (x - 2). When we multiply these binomials using the distributive property, we get x^2 + 3x - 10, which is the same as the given expression. Therefore, the correct answer is (x + 5)(x - 2).
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2.
Factor: x^{2} - 4x - 12
A.
(x + 4)(x - 3)
B.
(x - 4)(x + 3)
C.
(x + 6)(x - 2)
D.
(x - 6)(x + 2)
Correct Answer D. (x - 6)(x + 2)
Explanation The given expression is a quadratic trinomial in the form of x^2 - 4x - 12. To factorize it, we need to find two numbers that multiply to give -12 and add up to -4. The numbers -6 and 2 satisfy these conditions because (-6)(2) = -12 and (-6) + 2 = -4. Therefore, the correct answer is (x - 6)(x + 2).
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3.
Factor: x^{2} - 8x + 15
A.
(x - 3)(x - 5)
B.
(x + 3)(x + 5)
C.
(x - 3)(x + 5)
D.
(x + 3)(x - 5)
Correct Answer A. (x - 3)(x - 5)
Explanation The given expression is a quadratic trinomial in the form of ax^2 + bx + c. To factorize it, we need to find two numbers whose sum is equal to the coefficient of the middle term (-8) and whose product is equal to the constant term (15). The numbers that satisfy these conditions are -3 and -5. Therefore, the correct answer is (x - 3)(x - 5) since it represents the factored form of the given expression.
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4.
X^{2} - x - 2
A.
(x - 1)(x - 2)
B.
(x + 3)(x - 1)
C.
(x - 2)(x + 1)
D.
(x + 1)(x - 3)
Correct Answer C. (x - 2)(x + 1)
Explanation The given expression can be factored as (x - 2)(x + 1). This can be determined by using the distributive property to expand (x - 2)(x + 1) and simplifying the terms. The resulting expression is x^2 - x - 2, which matches the given expression. Therefore, (x - 2)(x + 1) is the correct answer.
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5.
X^{2 }+ x – 90
A.
(x – 9)(x – 10)
B.
(x – 9)(x + 10)
C.
(x + 9)(x – 10)
Correct Answer B. (x – 9)(x + 10)
Explanation The given expression is a quadratic trinomial that can be factored using the difference of squares formula. The correct answer, (x – 9)(x + 10), is obtained by multiplying the factors (x – 9) and (x + 10) together. This can be done by using the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. So, multiplying (x – 9) and (x + 10) gives x^2 + 10x - 9x - 90, which simplifies to x^2 + x - 90, the original expression.
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6.
X^{2} – 6x + 8
A.
(x – 4)(x + 2)
B.
(x + 4)(x + 2)
C.
(x – 4)(x – 2)
Correct Answer C. (x – 4)(x – 2)
Explanation The given expression can be factored as (x - 4)(x - 2) because when the expression is multiplied out, it results in x2 - 6x + 8. The first term in each factor is obtained by multiplying the x term in each factor, and the constant term in each factor (4 and 2) is obtained by multiplying the constant terms in each factor. The middle term in the expression (-6x) is obtained by multiplying the outer and inner terms in the factors (-4x and -2x), which gives -4x - 2x = -6x. Therefore, (x - 4)(x - 2) is the correct factorization.
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7.
Factor the trinomial below
x^{2} +7x + 6
A.
-6 and -1
B.
(x + 6)(x + 1)
C.
(x - 6)(x - 1)
D.
(x + 2)(x + 3)
Correct Answer B. (x + 6)(x + 1)
Explanation The given trinomial x^2 + 7x + 6 can be factored by finding two numbers that multiply to give 6 and add up to 7. The numbers -6 and -1 satisfy these conditions. Therefore, the factored form of the trinomial is (x + 6)(x + 1).
Explanation This is a trinomial. We want to multiply to -12 and add to a positive 1. Therefore we need +4 and -3.
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9.
2x^{2} + 2x – 4
A.
2(x – 1)(x + 2)
B.
2(x + 1)(x + 2)
C.
(2x – 1)(x + 4)
Correct Answer A. 2(x – 1)(x + 2)
Explanation The given expression is a quadratic trinomial. To factorize it, we can look for two binomials that multiply to give the trinomial. In this case, the binomials are (x - 1) and (x + 2). Multiplying these binomials together gives us 2(x - 1)(x + 2), which is the correct answer.
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10.
Factor: x^{2} + 8x + 15
A.
(x + 5)(x + 3)
B.
(x - 5)(x - 3)
C.
(x + 15)(x + 1)
D.
(x + 5)(x - 3)
Correct Answer A. (x + 5)(x + 3)
Explanation The given expression, x^2 + 8x + 15, can be factored as (x + 5)(x + 3). This is because when we multiply these factors, we get x^2 + 8x + 15. The first term of each factor, x, multiplied together gives x^2. The outer terms, x and 3, when multiplied give 3x. The inner terms, 5 and x, when multiplied give 5x. Finally, the last terms, 5 and 3, when multiplied give 15. Adding all these terms together gives us the original expression. Therefore, (x + 5)(x + 3) is the correct factorization.
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