Polynomial Functions and Rates of Change - AP Precalculus

$(x + 3)^2$. Complete the square or recognize perfect square trinomial.

-6. Expand and identify the constant term.

Constant polynomials. Non-zero constants have degree 0.

$x = 2, x = -2$. Set $f(x) = 0$ and solve for $x$.

$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$. Terms arranged by decreasing powers of $x$.

1. Set $x = 0$ and evaluate the function.

$7x^3$. The term containing $x$ to the third power.

1. The highest power of $x$ determines the degree.

$nax^{n-1}$. Power rule: bring down exponent, reduce power by 1.

1. The coefficient of the highest degree term.

As $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$. Odd degree with positive leading coefficient.

$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$. Terms arranged by decreasing powers of $x$.

$3x^2 + 4x - 1$. Apply power rule to each term.

1. The term with no variable (degree 0).

1. Evaluate $f'(2) = 2(2) = 4$ where $f'(x) = 2x$.

1. The exponent indicates how many times the root repeats.

$x^2 + x - 2$. Use FOIL to expand the product.

1. For $ax^2 + bx + c = 0$, sum of roots is $-\frac{b}{a}$.

Not possible; $a_n \neq 0$ for leading term. A polynomial must have a non-zero leading coefficient.

1. The highest power of $x$ is 5.

$x = 2, x = -2$. Set $f(x) = 0$ and solve for $x$.

$2x^4$. The term with the highest power of $x$.

$x = 2$ (multiplicity 2). Perfect square: $(x-2)^2 = 0$ gives repeated root.

1. The numerical factor multiplying $x^2$.

1. Product of three linear factors gives degree 3.

$15x^2 - 4x + 1$. Apply power rule to each term separately.

1. Coefficient of highest degree term when no coefficient shown is 1.

1. Evaluate $f'(1) = 6(1) - 4 = 2$ where $f'(x) = 6x - 4$.

$(x + 3)^2$. Complete the square or recognize perfect square trinomial.