3-5 Polynomials, Linear Factors, and Zeros

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, P(x) = 2x^3 - 4x^2 + 3x - 5 is a polynomial.

Front

What are linear factors of a polynomial?

Back

Linear factors are expressions of the form (x - r), where r is a root (zero) of the polynomial. A polynomial can be factored into linear factors if it can be expressed as a product of these factors.

Front

How do you find the zeros of a polynomial?

Back

To find the zeros of a polynomial, set the polynomial equal to zero and solve for the variable. For example, for P(x) = x^2 - 4, set x^2 - 4 = 0, which gives x = 2 and x = -2.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in P(x) = 3x^4 + 2x^3 - x + 7, the degree is 4.

Front

What is a cubic trinomial?

Back

A cubic trinomial is a polynomial of degree 3 with three terms. For example, P(x) = 2x^3 + 3x^2 - 5 is a cubic trinomial.

Front

What is the vertical stretch factor in a polynomial?

Back

The vertical stretch factor, often denoted as 'a' in the equation y = a * f(x), affects the steepness of the graph. If |a| > 1, the graph stretches vertically; if 0 < |a| < 1, it compresses vertically.

Front

How do you factor a polynomial?

Back

To factor a polynomial, look for common factors, use grouping, or apply special factoring techniques like difference of squares or perfect square trinomials. For example, x^2 - 9 can be factored as (x - 3)(x + 3).

Front

What is the relationship between the zeros and the factors of a polynomial?

Back

The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero. Each zero (r) gives rise to a linear factor (x - r).

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation with integer coefficients is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Front

What is synthetic division?

Back

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - r). It is faster than long division and is used to find the quotient and remainder.