Lesson 2.1: Polynomials, the Factor and Remainder Theorems

Introduction

Welcome to Lesson 2.1 of Foundation Further Mathematics! In this lesson, we will explore the fascinating world of polynomials. By the end of this lesson, you will be able to:

• Understand polynomial division and apply the remainder and factor theorems.
• Factorise cubic and quartic polynomials completely.
• Relate the coefficients of polynomials to their structure and behavior.
• Sketch polynomial graphs from their factored forms.
• Divide one polynomial by another and make use of the remainder theorem.

Hook: Why Should We Care About Polynomials?

Polynomials are everywhere! From calculating areas and volumes of shapes in geometry to modeling real-world scenarios like population growth or physics calculations, polynomials are an essential part of mathematics. Understanding them can help simplify complex problems and make calculations easier. Let’s dive into the world of polynomials and unlock their secrets! 🌟

What Are Polynomials?

A polynomial is a mathematical expression that consists of variables (like $x$) raised to whole number powers and coefficients (numbers that multiply these variables). The general form of a polynomial can be written as:

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

Where:

• $P(x)$ is the polynomial,
• $a_n, a_{n-1}, ..., a_1, a_0$ are constants (coefficients),
• $n$ is a non-negative integer representing the degree of the polynomial.

For example:

• $P(x) = 2x^3 - 3x^2 + 4x - 5$ is a polynomial of degree 3.
• $Q(x) = x^4 + 2$ is a polynomial of degree 4.

Polynomial Division: The Division Algorithm

Sometimes, we need to divide one polynomial by another. This process is similar to numerical long division. The division algorithm states that for any polynomials $f(x)$ and $g(x)$ (where g(x)

eq 0), we can express:

$$f(x) = g(x) \cdot q(x) + r(x)$$

Where:

• $q(x)$ is the quotient,
• $r(x)$ is the remainder and its degree is less than the degree of $g(x)$.

Example of Polynomial Division

Let’s divide $P(x) = 2x^3 - 3x^2 + 4x - 5$ by $g(x) = x - 2$.

1. To begin, divide the leading term of $P(x)$ by the leading term of $g(x)$:

$$\frac{2x^3}{x} = 2x^2$$

1. Multiply $g(x)$ by $2x^2$:

$$2x^2(x - 2) = 2x^3 - 4x^2$$

1. Subtract this from $P(x)$:

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

1. Repeat the process with the new polynomial until you reach a degree less than $g(x)$. The process will yield a final quotient and remainder.

The Remainder Theorem

The remainder theorem gives a powerful shortcut when evaluating polynomials. It states that if we divide a polynomial $P(x)$ by $x - c$, the remainder is simply $P(c)$. This can dramatically simplify our calculations!

Example of the Remainder Theorem

Using our previous polynomial $P(x) = 2x^3 - 3x^2 + 4x - 5$, if we want to find $P(2)$, we don’t need to perform long division:

$$P(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 = 2(8) - 3(4) + 8 - 5 = 16 - 12 + 8 - 5 = 7$$

So when dividing by $x - 2$, the remainder is 7. 📌

Factor Theorem

The factor theorem is a special case of the remainder theorem. It states that if $P(c) = 0$ for some $c$, then $(x - c)$ is a factor of $P(x)$. This means we can factor out $(x - c)$ from the polynomial!

Example of the Factor Theorem

Let’s check if $(x - 1)$ is a factor of $P(x) = 2x^3 - 3x^2 + 4x - 5$:

• Calculate $P(1)$:

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

$(x - 1)$ is not a factor since the remainder isn’t zero.

Now if we check $(x - 2)$:

• Calculate $P(2)$:

$$P(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 = 7$$

Still not a factor! But what if we try $(x - 3)$:

• Calculate $P(3)$:

$$P(3) = 2(3)^3 - 3(3)^2 + 4(3) - 5 = 0$$

Yes! $(x - 3)$ is a factor of $P(x)$. ✅

Sketching Polynomial Graphs

Once we have a polynomial in its factored form, we can use the zeros of the polynomial to sketch its graph. The x-intercepts occur at the values of $x$ for which $P(x) = 0$. Remember, if the factor $ (x - c) $ has an even power, the curve touches the x-axis and turns around. If the factor has an odd power, the curve crosses the x-axis.

Example

If we factor $P(x) = (x - 3)(2x^2 + 5)$:

• The x-intercept is at $x = 3$. Since $(x - 3)$ has an odd exponent, the graph crosses the x-axis.
• The term $2x^2 + 5$ is always positive (it has no real roots) so the graph moves upwards beyond the intercept. The overall shape will be U-shaped and cross the x-axis at $(3, 0)$.

Conclusion

In this lesson, we've learned how to divide polynomials and utilize the remainder and factor theorems effectively. Understanding these concepts is vital for exploring complex polynomial problems and graphing. Practice these skills, and you'll find your confidence in handling polynomials growing! 🌈

Study Notes

• A polynomial is expressed as $P(x) = a_n x^n + ... + a_0$.
• The division algorithm lets you express $f(x)$ as $g(x) \cdot q(x) + r(x)$.
• The remainder theorem states the remainder of $P(x)$ divided by $x - c$ is $P(c)$.
• The factor theorem indicates if $P(c) = 0$, then $(x - c)$ is a factor of $P(x)$.
• Sketch graphs using the x-intercepts from the factored form of the polynomial.