Polynomials - Introduction

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Q1. Which of the following is the standard form of a quadratic equation?

Solution:

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

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Q2. Solve for x: 5x + 2 = 17.

Solution:

Subtract 2 from both sides: 5x = 15. Then divide by 5: x = 3.

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Q3. Which of the following represents a quadratic equation?

Solution:

A quadratic equation is in the form ax^2 + bx + c = 0. The second option fits this form.

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Q4. What is the leading coefficient of the polynomial 4x^3 - 2x^2 + 7?

Solution:

The leading coefficient is the coefficient of the term with the highest degree. Here, it is 4.

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Q5. What is the solution set for the inequality 2x - 4 < 6?

Solution:

Add 4 to both sides: 2x < 10. Then divide by 2: x < 5.

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Q6. What is the solution to the inequality 3x - 5 < 4?

Solution:

Add 5 to both sides: 3x < 9. Then divide by 3: x < 3.

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Q7. What is the result of factoring the polynomial x^2 - 9?

Solution:

x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).

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Q8. What is the sum of the roots of the polynomial x^2 - 4x + 4?

Solution:

The sum of the roots of a quadratic equation ax^2 + bx + c is given by -b/a. Here, -(-4)/1 = 4.

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Q9. What is the product of the roots of the polynomial x^2 + 3x + 2?

Solution:

The product of the roots of a quadratic equation ax^2 + bx + c = 0 is given by c/a. Here, 2/1 = 2.

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Q10. What is the degree of the polynomial 3x^4 - 5x^2 + 2?

Solution:

The degree of a polynomial is the highest power of the variable. Here, the highest power is 4, so the degree is 4.