4.5 Solve Polynomials

Front

What are the zeros of a polynomial?

Back

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

Front

How do you find the zeros of the polynomial f(x) = x^4 + 8x^2 - 9?

Back

To find the zeros, set f(x) = 0 and solve for x. The zeros are 1, -1, ±3i.

Front

What does it mean to factor a polynomial completely?

Back

Factoring a polynomial completely means expressing it as a product of its irreducible factors.

Front

Factor the polynomial 2n^3 + 7n^2 - 2n - 7 completely.

Back

(n + 1)(n - 1)(2n + 7) is the complete factorization.

Front

What is the first step in factoring a polynomial?

Back

The first step is to look for a common factor in all the terms.

Front

Factor the polynomial 2x^3 + 5x^2 - 8x - 20 completely.

Back

(x - 2)(x + 2)(2x + 5) is the complete factorization.

Front

What is the significance of the degree of a polynomial?

Back

The degree of a polynomial indicates the highest power of the variable and determines the polynomial's behavior and number of zeros.

Front

How do you determine if a polynomial can be factored?

Back

A polynomial can often be factored if it can be expressed as a product of lower-degree polynomials or if it has rational roots.

Front

Factor the polynomial k^4 - 10k^2 - 39 completely.

Back

(k^2 - 13)(k^2 + 3) is the complete factorization.

Front

What is synthetic division and when is it used?

Back

Synthetic division is a simplified method of dividing a polynomial by a linear factor and is used when finding zeros.