Roots of Polynomials

Front

What are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of x for which the function equals zero. They are also known as the roots of the polynomial.

Front

How do you find the roots of the equation y = x^2 + 3x - 18?

Back

To find the roots, set the equation to zero: x^2 + 3x - 18 = 0. Factor or use the quadratic formula to find the roots, which are x = 3 and x = -6.

Front

What is a relative maximum in a graph?

Back

A relative maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.

Front

Identify the relative maximum of the function at the point (1, 4).

Back

The point (1, 4) is a relative maximum because it has a higher y-value than points immediately adjacent to it.

Front

Solve for the values of x in the equation (x - 5)(x - 1) = 0.

Back

The solutions are x = 5 and x = 1.

Front

What is the significance of the x-intercepts of a polynomial function?

Back

The x-intercepts (or roots) of a polynomial function indicate where the graph crosses the x-axis, representing the values of x for which the function equals zero.

Front

Explain the process of factoring a quadratic equation.

Back

To factor a quadratic equation, express it in the form ax^2 + bx + c = 0, and find two numbers that multiply to ac and add to b. Rewrite the equation as a product of two binomials.

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the roots of a quadratic equation.

Front

How can you determine the number of real roots of a polynomial?

Back

The number of real roots can be determined using the discriminant (b² - 4ac) for quadratics. If it's positive, there are two real roots; if zero, one real root; if negative, no real roots.

Front

What does it mean for a polynomial to be factored completely?

Back

A polynomial is factored completely when it is expressed as a product of irreducible polynomials over the real numbers.