graphing polynomials

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What is a polynomial?

Back

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

Front

What does it mean for a polynomial to be increasing on an interval?

Back

A polynomial is increasing on an interval if, for any two points x1 and x2 in that interval where x1 < x2, the value of the polynomial at x1 is less than the value at x2 (f(x1) < f(x2)).

Front

What are the zeros of a polynomial?

Back

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

Tags

CCSS.HSF-IF.C.7C

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 x. This may involve factoring, using the quadratic formula, or other algebraic methods.

Front

What is a leading coefficient?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the end behavior of the polynomial's graph.

Front

What does the degree of a polynomial indicate?

Back

The degree of a polynomial indicates the highest power of the variable in the polynomial. It affects the shape and number of turns in the graph.

Front

What is a y-intercept?

Back

The y-intercept of a polynomial is the point where the graph intersects the y-axis. It can be found by evaluating the polynomial at x = 0.

Front

How do you determine the y-intercept of f(x) = x^3 - 5x^2 - 25x + 125?

Back

To find the y-intercept, substitute x = 0 into the polynomial: f(0) = 0^3 - 5(0^2) - 25(0) + 125 = 125. Thus, the y-intercept is (0, 125).

Tags

CCSS.HSF-IF.C.7C

Front

What is the significance of the sign of the leading coefficient?

Back

The sign of the leading coefficient determines the direction of the ends of the polynomial's graph: if positive, the graph rises to the right; if negative, it falls to the right.

Front

What does it mean for a polynomial to have complex roots?

Back

Complex roots occur in conjugate pairs and indicate that the polynomial does not intersect the x-axis at those points. They arise when the discriminant of a quadratic is negative.

Tags

CCSS.HSN.CN.C.9