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# Chapter 2: Polynomials ## 2.1 Polynomial Functions ### Definitions **Polynomial:** An expression of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0$ where: * $x$ is variable * $n$ is a non-negative integer * $a_n, a_{n-1},..., a_1, a_0$ are coefficients (real numbers) **...

# Chapter 2: Polynomials ## 2.1 Polynomial Functions ### Definitions **Polynomial:** An expression of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0$ where: * $x$ is variable * $n$ is a non-negative integer * $a_n, a_{n-1},..., a_1, a_0$ are coefficients (real numbers) **Degree:** The highest power of $x$ (i.e., $n$) **Leading Coefficient:** $a_n$ **Constant Term:** $a_0$ ### Examples 1. $f(x) = 3x^2 + 2x + 1$ * Degree: 2 * Leading Coefficient: 3 * Constant Term: 1 2. $g(x) = x^3 - 5x$ * Degree: 3 * Leading coefficient: 1 * Constant Term: 0 3. $h(x) = 7$ * Degree: 0 * Leading Coefficient: 7 * Constant Term: 7 ### Non-Examples 1. $f(x) = x^{-1} + 2$ (negative exponent) 2. $g(x) = \sqrt{x}$ (fractional exponent) 3. $h(x) = \frac{1}{x}$ (variable in denominator) ### Graphing Polynomials * Polynomials are smooth, continuous curves. * The degree of the polynomial affects the shape of the graph. #### End Behavior The end behavior of a polynomial is determined by the leading term, $a_n x^n$. * If $n$ is even: * If $a_n > 0$, both ends point up. * If $a_n < 0$, both ends point down. * If $n$ is odd: * If $a_n > 0$, the left end points down, and the right end points up. * If $a_n < 0$, the left end points up, and the right end points down. ### Zeros (Roots) of Polynomials **Zero:** A value $x = c$ such that $f(c) = 0$. Graphically, a zero is an x-intercept. **Factor Theorem:** If $x = c$ is a zero of $f(x)$, then $(x - c)$ is a factor of $f(x)$. **Example:** $f(x) = (x - 2)(x + 1)(x - 3)$ Zeros: $x = 2, -1, 3$ ### Multiplicity If $(x - c)^k$ is a factor of $f(x)$, then $x = c$ is a zero with multiplicity $k$. * If $k$ is odd, the graph crosses the x-axis at $x = c$. * If $k$ is even, the graph touches (is tangent to) the x-axis at $x = c$. **Example:** $f(x) = (x - 1)^2 (x + 2)$ * $x = 1$ is a zero with multiplicity 2 (graph touches the x-axis) * $x = -2$ is a zero with multiplicity 1 (graph crosses the x-axis) ### Turning Points **Turning Point:** A point where the graph changes from increasing to decreasing or vice versa. * A polynomial of degree $n$ can have at most $n - 1$ turning points. ## 2.2 Dividing Polynomials ### Long Division Similar to dividing numbers, but with polynomials. **Example:** Divide $f(x) = x^3 - 3x^2 + 4x - 2$ by $g(x) = x - 1$ ``` x^2 - 2x + 2 x - 1 | x^3 - 3x^2 + 4x - 2 - (x^3 - x^2) ---------------- -2x^2 + 4x -(-2x^2 + 2x) ---------------- 2x - 2 -(2x - 2) ---------------- 0 ``` Quotient: $x^2 - 2x + 2$ Remainder: 0 $f(x) = (x - 1)(x^2 - 2x + 2)$ ### Synthetic Division A shortcut for dividing by a linear factor $(x - c)$. **Example:** Divide $f(x) = x^3 - 3x^2 + 4x - 2$ by $x - 1$ 1. Write down the coefficients of $f(x)$: $1, -3, 4, -2$ 2. Write down the value of $c$ (from $x - c$): $c = 1$ 3. Perform synthetic division: ``` 1 | 1 -3 4 -2 | 1 -2 2 ---------------- 1 -2 2 0 ``` Result: $x^2 - 2x + 2$ (quotient), 0 (remainder) ### Remainder Theorem If you divide $f(x)$ by $(x - c)$, the remainder is $f(c)$. **Example:** $f(x) = x^3 - 3x^2 + 4x - 2$ Divide by $(x - 1)$: Remainder is 0. $f(1) = (1)^3 - 3(1)^2 + 4(1) - 2 = 1 - 3 + 4 - 2 = 0$ ### Factor Theorem (Revisited) $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$ (i.e., the remainder is 0). ## 2.3 Zeros of Polynomial Functions ### Rational Root Theorem If $f(x) = a_n x^n +... + a_0$ has integer coefficients, then any rational zero of $f(x)$ must be of the form: $\frac{p}{q}$ where $p$ is a factor of $a_0$ (constant term) and $q$ is a factor of $a_n$ (leading coefficient). **Example:** $f(x) = 2x^3 + x^2 - 7x - 6$ Possible rational zeros: $\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}$ ### Finding Zeros 1. List possible rational zeros using the Rational Root Theorem. 2. Test these possible zeros using synthetic division or direct substitution. 3. If you find a zero, say $x = c$, then divide $f(x)$ by $(x - c)$ to get a quotient. 4. Repeat the process with the quotient or use the quadratic formula if the quotient is quadratic. **Example:** $f(x) = x^3 - 6x^2 + 11x - 6$ Possible rational zeros: $\pm 1, \pm 2, \pm 3, \pm 6$ Test $x = 1$: ``` 1 | 1 -6 11 -6 | 1 -5 6 ------------ 1 -5 6 0 ``` So, $x = 1$ is a zero, and $f(x) = (x - 1)(x^2 - 5x + 6)$ Factor the quadratic: $x^2 - 5x + 6 = (x - 2)(x - 3)$ Zeros: $x = 1, 2, 3$ ### Complex Zeros * Complex zeros occur in conjugate pairs. If $a + bi$ is a zero, then $a - bi$ is also a zero. **Example:** If $x = 2 + i$ is a zero, then $x = 2 - i$ is also a zero. ### Fundamental Theorem of Algebra A polynomial of degree $n$ has exactly $n$ complex zeros (counting multiplicities). ## 2.4 Real-World Applications * Polynomials can model various real-world situations, such as projectile motion, optimization problems, and curve fitting. **Example:** The height of a projectile can be modeled by a quadratic function: $h(t) = -16t^2 + v_0t + h_0$ where $t$ is time, $v_0$ is initial velocity, and $h_0$ is the initial height.