Lesson 1.4: Polynomials and Basic Division

Introduction

In this lesson, we will delve into polynomials and the basics of division. Understanding polynomials is crucial as they form the foundation of many areas in mathematics, including algebra, calculus, and more. By the end of this lesson, students will be able to:

• Add, subtract, and multiply polynomials.
• Divide a polynomial by a linear factor and identify the remainder.
• Sketch the shape of a simple factorized polynomial from its roots.
• Carry out arithmetic with polynomial expressions.
• State the quotient and remainder when dividing a polynomial by a linear factor.

Hook

Consider the expression $x^2 + 3x + 2$. This polynomial has roots at $x = -1$ and $x = -2$. What happens if we want to divide this polynomial by one of its linear factors, say $(x + 1)$? The result leads to a better understanding of polynomial behaviors and their graphs.

Understanding Polynomials

A polynomial is an algebraic expression comprised of variables and coefficients, combined using addition, subtraction, and multiplication, and raised to non-negative integer powers. The general form of a polynomial in one variable $x$ is:

$$ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 $$

where:

• $P(x)$ is the polynomial,
• $n$ is a non-negative integer representing the degree of the polynomial,
• $a_n, a_{n-1}, \ldots, a_0$ are coefficients, and
• $x$ is the variable.

Types of Polynomials

1. Monomial: A polynomial with only one term. Example: $5x^3$.
2. Binomial: A polynomial with two terms. Example: $x^2 + 3$.
3. Trinomial: A polynomial with three terms. Example: $x^2 + 3x + 2$.

Operations with Polynomials

Now, let’s discuss how to add, subtract, and multiply polynomials.

Adding Polynomials

To add two polynomials, combine like terms. For example, when adding $P(x) = 2x^2 + 3x + 4$ and $Q(x) = x^2 + 5$:

$$P(x) + Q(x) = (2x^2 + 3x + 4) + (x^2 + 5) = (2x^2 + x^2) + 3x + (4 + 5) = 3x^2 + 3x + 9$$

Subtracting Polynomials

Subtracting polynomials also involves combining like terms. For example, when subtracting $Q(x)$ from $P(x)$:

$$P(x) - Q(x) = (2x^2 + 3x + 4) - (x^2 + 5) = (2x^2 - x^2) + 3x + (4 - 5) = x^2 + 3x - 1$$

Multiplying Polynomials

Multiplication can be done using the distributive property or the FOIL method (for binomials). For instance, when multiplying $(x + 2)(x + 3)$:

$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Dividing a Polynomial by a Linear Factor

Division is also an important operation with polynomials. We will focus on dividing a polynomial by a linear factor.

Polynomial Long Division

Polynomial long division is a process similar to numerical long division. Let's divide $P(x) = x^3 + 3x^2 + 2x + 5$ by $D(x) = x + 1$:

1. Setup the long division.
2. Divide the leading term of the dividend by the leading term of the divisor: $\frac{x^3}{x} = x^2$.
3. Multiply the entire divisor by this result: $(x + 1)(x^2) = x^3 + x^2$.
4. Subtract the result from the original polynomial: $(x^3 + 3x^2 + 2x + 5) - (x^3 + x^2) = 2x^2 + 2x + 5$.
5. Repeat the process using the new polynomial:

• Divide $ \frac{2x^2}{x} = 2x$.
• Multiply: $2x(x + 1) = 2x^2 + 2x$.
• Subtract: $ (2x^2 + 2x + 5) - (2x^2 + 2x) = 5$.

1. Now we can summarize our division:

• The quotient is $x^2 + 2x$ and the remainder is $5$.

Therefore,

$$ P(x) = (x + 1)(x^2 + 2x) + 5 $$

Remainder Theorem

The Remainder Theorem states that when a polynomial $P(x)$ is divided by a linear divisor of the form $x - c$, the remainder is equal to $P(c)$. This provides a quick way to find the remainder without performing long division.

Example: Using the Remainder Theorem

For example, to find the remainder when dividing $P(x) = x^3 + 2x^2 + 3x + 4$ by $x - 2$, substitute $x = 2$:

$$ P(2) = 2^3 + 2(2^2) + 3(2) + 4 = 8 + 8 + 6 + 4 = 26 $$

Thus, the remainder is $26$.

Sketching the Shape of a Polynomial

Once we understand the roots of a polynomial, we can draw its graph. The polynomial formula will influence the shape of the graph. A polynomial of the form $P(x) = a_n (x - r_1)(x - r_2)...(x - r_n)$, where $r_i$ are the roots, will be a smooth curve that crosses the x-axis at the roots.

Example

Consider the polynomial $P(x) = (x + 1)(x - 2)(x - 3)$:

• Its roots are $x = -1$, $x = 2$, and $x = 3$.
• The graph will cross the x-axis at these points.
• For values of $x$ that are far away from the roots, the behavior of the graph will be dominated by the leading coefficient.

Graphing Tips

1. Identify the roots and plot them on the x-axis.
2. Determine the degree of the polynomial to understand how many times it will cross the axis (if it is even or odd).
3. Evaluate some key points to understand the behavior between roots.
4. Sketch the smooth curve connecting these points.

Conclusion

In this lesson, students learned the fundamental aspects of polynomials, including how to add, subtract, multiply, and divide them. The division of polynomials using long division and the Remainder Theorem was also covered. Understanding these concepts is essential for more advanced topics in mathematics. With these skills, students is equipped to handle polynomial expressions confidently.

Study Notes

• A polynomial is a sum of terms including variables raised to non-negative powers.
• Addition and subtraction of polynomials require combining like terms.
• Multiplication of polynomials can be performed using distribution.
• Polynomial long division is analogous to numerical long division.
• The Remainder Theorem allows for quick calculation of remainders in division.
• Polynomials can be visualized by understanding their roots and structure.