Given a polynomial f(x), if x – 4 is a factor, what else must be true?
If x – 4 is a factor of the polynomial f(x), then it must also be true that f(4) = 0. This follows from the Factor Theorem, which states that a polynomial f(x) has a factor of the form x – c if and only if f(c) = 0. In this case, because x – […]
What is the value of cot(θ) if sin(θ) = 8/9 in a right triangle?
To find the value of cot(θ) when sin(θ) = 8/9, we can use the relationship between the trigonometric functions. We know that cot(θ) is the reciprocal of tan(θ), and tan(θ) can be determined using the sine and cosine functions. Given that sin(θ) = 8/9, we can find cos(θ) using the Pythagorean identity: sin²(θ) + cos²(θ) […]
Is there a number that is exactly 5 more than its cube?
Let’s explore the equation that describes this problem. We are trying to find a number, say x, such that: x = x3 + 5 This can be rearranged to: x3 – x + 5 = 0 Now, the next step is to check if there are any real solutions to this cubic equation. We can […]
Blake knows that one of the solutions to x² – 6x + 8 = 0 is x = 2. What is the other solution?
To find the other solution to the quadratic equation x² – 6x + 8 = 0, we can use the fact that if x = 2 is one solution, we can factor the quadratic equation. The factored form of a quadratic equation is expressed as (x – r₁)(x – r₂) = 0, where r₁ and […]
Describe the graph of y = 34x + 12 as compared to the graph of y = x
The graph of the equation y = 34x + 12 is a straight line with a much steeper slope compared to the graph of y = x. First, let’s break down both equations: y = x: This is a simple linear equation where the slope is 1. This means for every unit increase in x, […]
If sin a = 1/10 and cos b = 2/29, find tan(2a) where a and b lie between 0 to π/4.
To find tan(2a), we can use the double angle formula for tangent: tan(2a) = 2tan(a) / (1 – tan2(a)) First, we need to find tan(a). We know: sin(a) = 1/10 Now, we can calculate cos(a) using the Pythagorean identity: cos(a) = √(1 – sin2(a)) Calculating that: cos(a) = √(1 – (1/10)2) = √(1 – 1/100) […]
What is the quotient of 125, 8, 3, 25, 10, and 4?
To find the quotient of the numbers 125, 8, 3, 25, 10, and 4, we first need to clarify how we want to divide them. Quotient usually refers to the result of division. Let’s divide them step by step. We can start by dividing 125 by 8: 125 ÷ 8 = 15.625 Next, we can […]
How to Find the X-Intercepts of a Parabola with Vertex and Y-Intercept?
To find the x-intercepts of a parabola, we first need to establish its equation based on the given vertex and y-intercept. The vertex form of a parabola is given by: y = a(x – h)² + k Here, (h, k) is the vertex of the parabola. From the question, we know that the vertex is […]
What is the formula for surface area of a cylinder?
The formula for the surface area of a cylinder is given by: A = 2πr(h + r) Where: A is the surface area r is the radius of the base of the cylinder h is the height of the cylinder To break it down, a cylinder has two circular bases and a curved surface that […]
How do I write the equation 5x + 2y = 10 in the form y = mx + b?
To rewrite the equation 5x + 2y = 10 in the slope-intercept form, which is y = mx + b, we need to isolate y on one side of the equation. Here are the steps to follow: Start with the original equation: 5x + 2y = 10. Subtract 5x from both sides to get 2y […]