(Grade 11) Section 2.2 - Factoring Polynomials: Common Factoring

Explanation
The greatest common factor (GCF) is the largest number or term that divides evenly into all the terms of a polynomial. In this case, the GCF of 8x, 14x, and -32 is 2, because it is the largest number that can divide evenly into all three terms. Therefore, the correct answer is 2.

Explanation
The greatest common factor (GCF) is the largest number or term that divides evenly into all the terms of a given expression. In this case, the GCF of the terms -24x^2 and -18x is -6x. This is because -6x is the largest term that can be factored out from both terms, resulting in -6x(4x + 3). Therefore, -6x is the correct answer.

Explanation
The given expression, 35x^2 - 14x, can be factored by taking out the greatest common factor, which is 7x. This leaves us with 7x(5x - 2). This is the fully factored form of the polynomial.

Explanation
The given expression can be factored by using the distributive property. We can factor out the common binomial (4x - 3) from both terms in the expression. This results in (4x - 3) multiplied by (2x + 5), which gives us the fully factored form of the polynomial. Therefore, the correct answer is (4x - 3)(2x - 5).

Explanation
The missing factor in the equation 12b^2 - 72b = 12(b - 6) is b - 6. This can be determined by factoring out the common factor of 12 from the equation, resulting in 12(b^2 - 6b) - 12(6) = 0. Then, dividing both sides by 12 gives b^2 - 6b - 6 = 0, which can be further simplified to (b - 6)(b + 1) = 0. Therefore, the missing factor is b - 6.

Explanation
The area of a parallelogram is equal to the base multiplied by the height. In this case, the area is given as 25x^2 + 75x and the height is x + 3. We can set up the equation (x + 3) * base = 25x^2 + 75x. To find the measure of the base, we need to solve for base. By dividing both sides of the equation by (x + 3), we get base = (25x^2 + 75x) / (x + 3), which simplifies to base = 25x. Therefore, the measure of the base is 25x.

Explanation
The given expression 25x^2 + 9 represents the area of the square. To find the measure of one side of the square, we need to find the square root of the area. Taking the square root of 25x^2 + 9 gives us √(25x^2 + 9). However, this cannot be simplified further, so the measure of one side of the square is √(25x^2 + 9). Therefore, none of the given options (x+3, 5x+1, 5x+3, 5x+9) is the correct answer.

Explanation
The area of the label is given by the expression 10n^2 + 20n - 5. To find the length of the label, we need to find the value of n that makes this expression equal to 5. By comparing the given answer choices with the expression, we can see that the expression 2n^2 + 4n - 1 matches the given expression. Therefore, the length of the label is given by 2n^2 + 4n - 1.

Explanation
The surface area of a sphere is given by the formula SA = 4πr^2. In this question, the surface area of the sphere is given as 4πx^2 – 8πx + 4π. To find the measure of the radius, we need to equate the given surface area equation to the formula for surface area and solve for r. Equating the two equations, we get 4πr^2 = 4πx^2 – 8πx + 4π. Simplifying this equation, we get r^2 = x^2 – 2x + 1. Taking the square root of both sides, we get r = x – 1. Therefore, the measure of the radius is x – 1.