Rational Zeros Theorem Quiz

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Questions and Answers

What information about a parabola can be determined from its vertex?

The vertex gives the maximum or minimum point of the parabola and indicates whether it opens upward or downward.

How do you determine the x-intercepts of a quadratic function?

The x-intercepts can be found by setting the quadratic equation to zero and solving for x.

What does the axis of symmetry of a parabola represent?

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves.

What is the standard form of a quadratic function?

The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. Signup and view all the answers

Describe how to find the maximum or minimum value of a quadratic function.

The maximum or minimum value can be found by evaluating the function at the vertex, given by $x = -\frac{b}{2a}$. Signup and view all the answers

How does the leading coefficient of a quadratic function affect the graph?

If the leading coefficient is positive, the parabola opens upward; if negative, it opens downward. Signup and view all the answers

What role does the discriminant play in finding the roots of a quadratic equation?

The discriminant, found using $b^2 - 4ac$, determines the number and nature of the roots of the quadratic equation. Signup and view all the answers

How do you convert a quadratic function from vertex form to standard form?

To convert, you expand the expression $a(x - h)^2 + k$ into the form $ax^2 + bx + c$. Signup and view all the answers

List the multiplicities of the zeros for the polynomial function $f(x) = 6x(x-13)(x+2)^3$.

Zero 0 has multiplicity 1, zero 13 has multiplicity 1, and zero -2 has multiplicity 3. Signup and view all the answers

What is the factored form of a fourth-degree polynomial having zeros at -4, 2, 3, and -5?

The polynomial is $f(x) = (x+4)(x-2)(x-3)(x+5)$. Signup and view all the answers

Identify the $y$-intercept and all $x$-intercepts of the function $f(x) = 2x^3 + 2x^2 - 24x$.

The $y$-intercept is (0, 0) and the $x$-intercepts are (-4, 0), (0, 0), and (3, 0). Signup and view all the answers

Describe the end behavior of the polynomial function $f(x) = 6x^6 - 3x^4 - 4x^3 + 6x^2$.

The function rises to the left and rises to the right. Signup and view all the answers

For the inequality $(x+4)(x-7) ≥ 0$, what is the correct solution set?

The solution set is $x ≤ -4$ or $x ≥ 7$. Signup and view all the answers

What are the results of the division $(6x^3 + 9x^2 + 21x + 11) ÷ (3x^2 + 3x)$?

The quotient is $2x + 1$ and the remainder is $18x + 11$. Signup and view all the answers

What is the quotient and remainder when using synthetic division on $-4x^3 + 12x^2 + 18x - 14x + 24$ divided by $x - 4$?

The quotient is $-4x^2 + 4x + 2$ and the remainder is 0. Signup and view all the answers

What is the leading coefficient of the polynomial $-10 + 18x^6 + 4x^3$ and what is its degree?

The leading coefficient is 18 and the degree is 6. Signup and view all the answers

What intervals identify where the function increases based on its local extrema?

The function increases on the intervals $(-∞, -8)$, $(-5, -2)$, $(2, 5)$, and $(5, ∞)$. Signup and view all the answers

Simplify the expression $rac{6x^2 + 11x}{2x}$ and provide the result.

The simplified answer is $3x + rac{11}{2}$. Signup and view all the answers

From the polynomial function $f(x) = -5x^3 + 4x^2 - 8x - 1$, what is its end behavior?

It rises to the left and falls to the right. Signup and view all the answers

Using the remainder theorem, what is the value of $P(2)$ for the polynomial $P(x) = x^3 - 2x^2 - 4$?

The remainder is -4, hence $P(2) = -4$. Signup and view all the answers

Identify the signs of the leading coefficient and suggest a possible degree for the function with positive leading coefficient.

The leading coefficient is positive and a possible degree is 6 or 8. Signup and view all the answers

For the quadratic $y = -3x^2$, which direction does the parabola open?

The parabola opens downward. Signup and view all the answers

Find the real zeros of the function $f(x) = -5x(x^2 - 16)(x^2 + 36)$.

The real zeros are $x = -4, 0, 4$. Signup and view all the answers

In the polynomial function $f(x) = 4x(x-4)(x+2)$, what is the end behavior?

It falls to the left and rises to the right. Signup and view all the answers

Study Notes