Okay, let's delve deeper into polynomials, focusing on their roots and division. Roots (or Zeros) of Polynomials The roots or zeros of a polynomial $P(x)$ are the values of $x$ for which $P(x) = 0$. These are the x-intercepts when the polynomial is graphed. • Factor Theorem: If $x=c$ is a root of a polynomial $P(x)$, then $(x-c)$ is a factor of $P(x)$. Conversely, if $(x-c)$ is a factor of $P(x)$, then $x=c$ is a root. Example: If $P(x) = x^2 - 4$, then $P(2) = 2^2 - 4 = 0$. So, $x=2$ is a root, and $(x-2)$ is a factor. Indeed, $x^2 - 4 = (x-2)(x+2)$. • Remainder Theorem: When a polynomial $P(x)$ is divided by $(x-c)$, the remainder is $P(c)$. Example: If $P(x) = x^2 + 3x + 1$ is divided by $(x-1)$, the remainder is $P(1) = 1^2 + 3(1) + 1 = 1 + 3 + 1 = 5$. • Rational Root Theorem: For a polynomial with integer coefficients, any rational root $\frac{p}{q}$ (in simplest form) must have $p$ as a factor of the constant term ($a_0$) and $q$ as a factor of the leading coefficient ($a_n$). This helps find possible rational roots to test. Polynomial Division Polynomial division is used to divide one polynomial by another. 1. Polynomial Long Division: Similar to numerical long division. Example: Divide $P(x) = x^3 - 2x^2 - 4x + 5$ by $D(x) = x - 1$. ` x^2 - x - 5 ________________ x - 1 | x^3 - 2x^2 - 4x + 5 (x^3 - x^2) ___________ -x^2 - 4x (-x^2 + x) ___________ -5x + 5 (-5x + 5) ___________ 0 ` So, $\frac{x^3 - 2x^2 - 4x + 5}{x - 1} = x^2 - x - 5$. The remainder is 0, which means $(x-1)$ is a factor and $x=1$ is a root. 2. Synthetic Division: A shortcut for dividing a polynomial by a linear factor of the form $(x-c)$. Example: Divide $P(x) = x^3 - 2x^2 - 4x + 5$ by $(x-1)$. Here, $c=1$. Write down the coefficients of the polynomial: $1 \quad -2 \quad -4 \quad 5$. Bring down the first coefficient. Multiply by $c$ and add to the next coefficient. Repeat. ` 1 | 1 -2 -4 5 | 1 -1 -5 ------------------ 1 -1 -5 0 ` The last number (0) is the remainder. The other numbers ($1 \quad -1 \quad -5$) are the coefficients of the quotient, starting with a degree one less than the original polynomial. So, the quotient is $1x^2 - 1x - 5 = x^2 - x - 5$. Graphing Polynomials (Key Features) • End Behavior: Describes what happens to the $y$-values of the polynomial as $x$ approaches positive or negative infinity. It's determined by the leading term ($a_n x^n$). If $n$ is even*: $a_n > 0$: Both ends go up (like $y=x^2$). $a_n < 0$: Both ends go down (like $y=-x^2$). If $n$ is odd*: $a_n > 0$: Left end goes down, right end goes up (like $y=x^3$). $a_n < 0$: Left end goes up, right end goes down (like $y=-x^3$). • Multiplicity of Roots: How many times a root appears. If a root $c$ has an odd multiplicity (e.g., $(x-c)^1$, $(x-c)^3$), the graph crosses* the x-axis at $x=c$. If a root $c$ has an even multiplicity (e.g., $(x-c)^2$, $(x-c)^4$), the graph touches* the x-axis at $x=c$ and turns around (like a bounce). • Turning Points: A polynomial of degree $n$ has at most $n-1$ turning points (local maxima or minima).