If a polynomial function f(x) has roots 3 and √7, what must also be a root of f(x)?
In the case of polynomial functions with real coefficients, complex roots and irrational roots come in conjugate pairs. If a polynomial has a root that is an irrational number, like √7, then its conjugate must also be a root of the polynomial. The conjugate of √7 is -√7. Therefore, if the polynomial function f(x) has […]
When x = 3 and y = 5, by how much does the value of 3x^2 + 2y exceed the value of 2x^2 + 3y?
To find out by how much the value of 3×2 + 2y exceeds 2×2 + 3y, we first need to substitute the values of x and y into these expressions. Given that x = 3 and y = 5: First, we calculate 3×2 + 2y: 3(32) + 2(5) = 3(9) + 10 = 27 + […]
How do I write the equation 3x² + 19x + 14 in standard form and factor the left side?
To write the equation in standard form, we need to ensure it is arranged as ax² + bx + c = 0. The given equation is: 3x² + 19x + 14 = 0 Here, the equation is already in standard form where a = 3, b = 19, and c = 14. Next, we will […]
Find the Total Surface Area of Cubes Having the Following Sides: 1 cm, 3 cm, 5 cm, and 55 cm
To find the total surface area of a cube, we can use the formula: Total Surface Area = 6 × a² where a is the length of one side of the cube. Now, let’s calculate the total surface area for each cube with the given side lengths: 1. For a cube with side 1 cm: […]
Which phrase matches the algebraic expression below 3x + 7 – 10?
The algebraic expression given is 3x + 7 – 10. To understand this expression, we need to simplify it first. By combining the constant terms, we can simplify 7 – 10 to get -3. So, the expression can be written as: 3x – 3 This means that the phrase that matches the algebraic expression describes […]
How Much Fabric Will Erica Have Left After Making a Curtain?
Erica bought three twelve yards of fabric, which means she has a total of: 3 x 12 = 36 yards of fabric If she uses 23 yards of that fabric to make a curtain, we can find out how much she has left by subtracting the amount used from the total she had: 36 – […]
Solve this quadratic equation using the quadratic formula: x² – 8x – 5 = 0
To solve the quadratic equation x² – 8x – 5 = 0 using the quadratic formula, we start by identifying the coefficients a, b, and c in the standard form ax² + bx + c = 0. Here, we have: a = 1 b = -8 c = -5 The quadratic formula is: x = […]
What is the area of a sector of a circle when the length of the arc is 4 cm and the radius is 6 cm?
To find the area of the sector of a circle given the length of the arc and the radius, we can use the following formula: Area of the sector
Factor 7x³ + 21x² + 3x + 9 by Grouping: What is the Resulting Expression?
To factor the expression 7x³ + 21x² + 3x + 9 by grouping, we start by grouping the terms into two pairs. We can group the first two terms together and the last two terms together: (7x³ + 21x²) + (3x + 9) Now, we factor out the greatest common factor from each group: From […]
Rewrite the Radical as a Rational Exponent: The Fourth Root of 7 to the Fifth Power
To rewrite the expression The Fourth Root of 7 to the Fifth Power, we can break it down step-by-step. The fourth root of any number can be expressed using a rational exponent as follows: If we take the fourth root of a number, say x, we can write it as: x1/4 In this situation, we’re […]