1.
What is the missing factor? x^{2} + 5x – 14 = (x +7)(?)
Correct Answer
D. X–2
Explanation
The correct missing factor is x-2. This can be determined by factoring the quadratic equation x^2 + 5x - 14 using the method of factoring by grouping or by using the quadratic formula. The factors of -14 that add up to 5 are -2 and 7, so the factored form of the equation is (x + 7)(x - 2). Therefore, the missing factor is x - 2.
2.
Which shows the factorization of the polynomial? w^{2}–5w+6
Correct Answer
C. (w–2)(w–3)
Explanation
The correct answer is (w–2)(w–3). This is the factorization of the given polynomial w^2–5w+6. By multiplying (w–2) and (w–3) together using the distributive property, we get w^2–2w–3w+6, which simplifies to w^2–5w+6. Therefore, (w–2)(w–3) is the correct factorization.
3.
What is the missing factor? –7y + y^{2} – 18 = (?)(y+2)
Correct Answer
D. Y–9
Explanation
In the equation -7y + y^2 - 18 = ( ? )(y+2), the missing factor can be determined by comparing the given equation with the answer choices. By analyzing the equation, it can be observed that the missing factor should be y-9. This is because when y-9 is multiplied by (y+2), it will result in -7y + y^2 - 18, which is the same as the left side of the equation. Therefore, y-9 is the missing factor that completes the equation.
4.
Factor the quadratic equation fully. –2x^{2} – 12x – 16
Correct Answer
B. –2(x+2)(x+4)
Explanation
The given quadratic equation is -2x^2 - 12x - 16. To factor it fully, we can find the common factors of the terms. The common factor here is -2. Factoring out -2, we get -2(x^2 + 6x + 8). Now, we need to factor the quadratic expression inside the parentheses. The factors of 8 that add up to 6 are 2 and 4. So, we can write the quadratic expression as (x + 2)(x + 4). Putting it all together, the fully factored form of the quadratic equation is -2(x + 2)(x + 4).
5.
The area of the rectangle is x^{2}+8x+12. If the length is x+2, what is the width?
Correct Answer
C. X+6
Explanation
The area of a rectangle is found by multiplying its length by its width. In this case, the area is given as x^2 + 8x + 12. We are also given that the length is x+2. To find the width, we can divide the area by the length. When we divide x^2 + 8x + 12 by x+2, we get x+6. Therefore, the width of the rectangle is x+6.
6.
The formula for the surface area of a cube is SA=6e^{2}. If the surface area of this cube is 6x^{2}–24x–24, what is the measure of one edge?
Correct Answer
A. X–2
Explanation
The given surface area of the cube is 6x^2 - 24x - 24. Comparing this with the formula SA = 6e^2, we can equate the expressions: 6x^2 - 24x - 24 = 6e^2. Simplifying this equation, we get x^2 - 4x - 4 = e^2. Since e represents the measure of one edge, the measure of one edge is equal to x - 2. Therefore, the correct answer is x - 2.
7.
Jasmine painted this design using black and white paint. She used (x^{2}+x–6)/2 cm^{2} of red paint. What are the dimensions of the black triangle?
Correct Answer
C. B=x–2;h=x+3
Explanation
The dimensions of the black triangle are b=x-2 and h=x+3.
8.
Which trinomial does not have (x–2) as a factor?
Correct Answer
D. X^2–2x–8
Explanation
The trinomial x^2–2x–8 does not have (x–2) as a factor because when we factor out (x–2) from the trinomial, we get (x–4)(x+2), not (x–2)(x+4). Therefore, x^2–2x–8 is the correct answer.
9.
Which value for k will allow this polynomial to be factored? x^{2}+kx+18
Correct Answer
D. 11
Explanation
To factor a polynomial, we need to find two binomials that multiply together to give us the original polynomial. In this case, we have x^2 + kx + 18. The coefficient of x^2 is 1, so the binomials will have the form (x + a)(x + b). To find the values of a and b, we need to find two numbers that add up to k and multiply to 18. The only pair of numbers that satisfies this condition is 9 and 2. Therefore, the polynomial can be factored as (x + 9)(x + 2). To make this possible, k must be equal to 11.
10.
Factor completely: k^{2} -2k - 24
Correct Answer
A. (k+4)(k-6)
Explanation
The given quadratic expression can be factored as (k+4)(k-6). This can be determined by finding two numbers whose product is -24 and whose sum is -2. The numbers -4 and 6 satisfy these conditions, so the expression can be written as (k-6)(k+4).