Unit 3 Honors Math 3: Review of 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 can be found by solving the equation p(x) = 0.

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What is multiplicity in relation to polynomial zeros?

Back

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. A zero with an even multiplicity will touch the x-axis, while a zero with an odd multiplicity will cross the x-axis.

Front

How do you find the zeros of the polynomial g(x) = (x+4)^2(x+8)^3(x-5)?

Back

Set g(x) = 0 and solve for x: The zeros are x = -4 (multiplicity 2), x = -8 (multiplicity 3), and x = 5 (multiplicity 1).

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions of the quadratic equation ax² + bx + c = 0.

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What are complex roots?

Back

Complex roots are solutions to polynomial equations that include imaginary numbers. They occur in conjugate pairs when coefficients are real.

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How do you determine the intervals where a polynomial is increasing or decreasing?

Back

To find increasing or decreasing intervals, analyze the first derivative of the polynomial. Increasing intervals occur where the derivative is positive, and decreasing intervals occur where the derivative is negative.

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What does it mean for a polynomial to have a zero with a wave behavior?

Back

A zero with wave behavior occurs when the multiplicity of the zero is odd, causing the graph to cross the x-axis at that point.

Front

How do you find the decreasing intervals of a polynomial?

Back

To find decreasing intervals, calculate the first derivative of the polynomial, set it to zero to find critical points, and test intervals around those points to see where the derivative is negative.

Front

What is the significance of the intervals where f(x) > 0?

Back

The intervals where f(x) > 0 indicate where the graph of the polynomial is above the x-axis, meaning the function has positive values.

Front

How do you solve a quadratic equation like x² - 5x + 10 = 0?

Back

Use the quadratic formula: x = (5 ± √(25 - 40)) / 2 = (5 ± i√15) / 2.