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Fundamental Theorem of Algebra

In this lesson, we will be discussing the Fundamental Theorem of Algebra. The idea behind this concept is mainly to factorize and solve polynomials by identifying the roots of a given expression. Before we begin, let us review the following definitions.

Multiplicity

If a polynomial p(x) has multiple roots at r, the multiplicity of r refers to the number of times (x – r) occurs as a factor of p(x). These are also called repeated roots. For example, p(x) = (x – r)3 means that the root r has a multiplicity of 3.

Roots

The roots of a polynomial p(x) are values of a variable that satisfy the equation p(x) = 0. These are also known as solutions, zeros, and x-intercepts.

In addition, let us also define the standard form and degree of a polynomial as below:

Let p(x) be a polynomial of the form

where are the coefficients to the variables respectively.

The degree of a polynomial is the highest power of x in a polynomial with non-zero coefficients.

With that in mind, we can establish the following theorem.

The Fundamental Theorem of Algebra

If p(x) is a polynomial of degree n ≥ 1, then p(x) = 0 has exactly n roots, including multiplicities and complex roots.

Note that n refers to the highest degree of a given polynomial. Proving this theorem is beyond the scope of this syllabus. Thus, it is not necessary for you to verify it! However, it is important that you know how to apply this concept to factoring and solving polynomials.

It is helpful to recall that the term complex here describes a complex root with a non-zero imaginary part, say, a + bi, where a is real and b ≠ 0. As complex roots always come in conjugate pairs, this implies that a - bi is also a root to the polynomial.

Application of the Fundamental Theorem of Algebra

Now, let us apply the Fundamental Theorem of Algebra. We shall present several worked examples in this segment that will provide us with a clearer understanding of the concept with regard to factoring and solving polynomials. For simplicity, we shall use the abbreviation FTA to define the Fundamental Theorem of Algebra.

Identifying the Number of Roots in a Polynomial

Using FTA, determine the number of roots for the polynomial

Solution

The degree of f(x) is n = 4, Thus, by FTA we have 4 solutions.

Using FTA, determine the number of roots for the polynomial

Solution

The degree of f(x) is n = 7, Thus, by FTA we have 7 solutions.

Therefore, from FTA we can easily deduce that:

• a linear polynomial (degree 1) will have one root

• a quadratic polynomial (degree 2) will have two roots

• a cubic polynomial (degree 3) will have three roots

• an nth degree polynomial will have n roots

Identifying the Zeros and Degree of an Equation

Let us first define a specific form of a factorized polynomial as below:

A polynomial p(x) of the form

can be rewritten as a product of linear factors as

where are the roots of the polynomial.

Given the factored polynomial below,

determine the number of roots each equation has and find all their solutions.

Solution

Setting f(x) = 0 and using the Zero Product Property, we find that the roots are

As there are 3 roots in total, so by FTA, the polynomial must be a degree of 3.

Given the factored polynomial below,

determine the number of roots each equation has and find all their solutions.

Solution

Similarly, as before, we find that the zeros of the polynomial are

As there are 6 zeros in total, so by FTA, the polynomial must be a degree of 6.

Previously, we had mentioned that complex roots always come in conjugate pairs. This means that polynomials of even degrees can be made up of either all real roots or all pairs of complex roots (or a combination of both). Polynomials of odd degrees, however, cannot be made up of all pairs of complex roots. In this case, it will be made up of a combination of real roots and pairs of complex roots (or only real roots). We shall demonstrate this with the following examples.

Factorize and solve the polynomial below.

Solution

We first set f(x) = 0 as

Observe that this is a difference of two squares. From Special Products, we know that this becomes

Similarly, we can further factorize as

Solving this, we obtain

This polynomial has an even degree of 4. This, we have 4 roots made up of 2 real roots and 2 complex conjugate roots.

Factorize and solve the polynomial below.

Solution

Setting f(x) = 0, we have

Using the grouping method, we can factorize this as

Solving this we have

This polynomial has an odd degree of 3. Thus we obtain 3 roots made up of one real root and 2 complex conjugate roots.

So far, we have looked at polynomials that can be factorized as a product of linear factors. In some cases, we may encounter irreducible quadratics. An irreducible quadratic is one that we can no longer break down into a product of linear factors.

Take the last two examples for instance. The expressions x2 + 9 and x2 + 4 are examples of irreducible quadratics. We find that these expressions are made up of a product of 2 complex conjugates roots. Here,

Multiplying a pair of complex conjugate roots takes the general formula:

This brings us to the question: what if the irreducible quadratic is not of the form above? We thus need to find a method to identify an irreducible quadratic. To do this, we shall make use of the discriminant of a quadratic polynomial. The following is a general rule we should follow when we encounter such quadratics.

The Discriminant of a Quadratic Equation

the discriminant describes the roots of the polynomial. There are three cases to consider here.

Case 1: D > 0

p(x) can be reduced further into a product of linear factors. We will obtain two distinct real roots.

Case 2: D = 0

p(x) can be reduced further into a product of multiplicities. We will obtain one real repeated root.

Case 3: D < 0

p(x) becomes an irreducible quadratic. We will obtain two complex conjugate roots.

Factorize and solve the polynomial below.

Solution

Setting f(x) = 0, we have .

By FTA, we observe that f(x) has a degree of n = 3, and so we must have 3 solutions. Using long division, we can factorize f(x) as .

From here, we can see that the equation is an irreducible quadratic since the discriminant is less than zero as below.

Thus, we must use the quadratic formula to solve the remaining two roots as

Therefore, we have one real root, x = 2 and a pair of complex conjugate roots

Building a Polynomial Using FTA

In this final section, we will show two worked examples that will demonstrate how we can use FTA to create a polynomial from a given statement.

Write an algebraic expression in the standard polynomial form where the zeros are 3 and –5. The polynomial has a degree of 3 and the root –5 has a multiplicity of 2.

Solution

If the zeros of the polynomial, say f(x) are 3 and –5, then f(x) will have factors of (x – 3) and (x + 5).

We also know that the degree of f(x) is 3, so by FTA, we must have 3 roots or in other words, 3 factors. So f(x) would look something like

We also know that the multiplicity of -5 is 2, therefore, we must have two factors of (x + 5). Thus, we obtain

Expanding this using the FOIL method, we have the standard form of f(x) as

Write an expression in the standard polynomial form where the zeros are –3, –4i, and 4i. The root –3 has a multiplicity of 2. What is the degree of this polynomial?

Solution

Since -4i and 4i are a pair of complex conjugate roots, we may use the product of two complex conjugate pairs. If the zeros of the polynomial, say f(x) are –3, –4i, and 4i, then f(x) will have factors of (x + 3) and (x2 + 16).

We also know that the multiplicity of –3 is 2, therefore, we must have two factors of (x + 3). Thus, we obtain the completely factored form below.

From the statement and the factored form above, we find that f(x) contains 4 roots: 2 real repeated roots, x = –3, and a pair of complex conjugate roots, x = –4i and x = 4i. So, by FTA, f(x) must be a degree of 4.

Expanding this using the FOIL method, we have the standard form of f(x) as

Fundamental Theorem of Algebra - Key takeaways

• The Fundamental Theorem of Algebra states that a polynomial p(x) of degree n has n roots when p(x) = 0.
• A polynomial of a the form p(x) = an xn + ... + a1 x1 + a0 , can be factorized as a product of linear factors of the form p(x) = a( x – r1 )( x – r2 ) ... ( x – rn ).
• The zeros of a polynomial may be in the form of real numbers, multiplicities, or complex numbers.
• Complex numbers always come as a pair of complex conjugates.
• A polynomial can be factored into a product of the following two forms:
1. a linear factor
• A multiplicity is a repeated root, that is a factor that appears more than once in an expression.

The fundamental theorem of algebra states a polynomial of degree n has n roots.

We apply the fundamental theorem of algebra to factoring and solving polynomials.

The fundamental theorem of linear algebra states that every polynomial has at least one zero in the complex numbers that can be represented as a linear factor.

The roots of a polynomial are values of a variable that satisfy the equation. These are also known as solutions, zeros and x-intercepts.

We conduct the fundamental theorem of algebra by identifying the degree of the polynomial to determine the number of roots the polynomial has.

Final Fundamental Theorem of Algebra Quiz

Question

How can we tell if a line will be dotted or solid for an inequality?

If the inequality is just less than or greater than, then the line is dotted. If we have $$\le$$ or $$\ge$$ then we are looking at a solid line as the inequality can also be equal to.

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Question

What does the variable k govern in the turning point formula?

Vertical translation of the graph and the y-coordinate of the turning point vertex.

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Question

What does the variable h govern in the turning point formula?

Horizontal translation of the graph and the x-coordinate of the turning point vertex.

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Question

What does the variable $$a$$ govern in the turning point formula?

concavity and dilation

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Question

What is multiplicity? What is another word for this?

If a polynomial p(x) has multiple roots at r, the multiplicity of r refers to the number of times (x - r) occurs as a factor of p(x). These are also called repeated roots.

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Question

What is the purpose of the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra is used to factorize and solve polynomials by identifying the roots of a given expression.

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Question

What is a linear factor?

A linear factor is a factor of a polynomial of degree 1. This may take the form (ax + b).

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Question

A polynomial can be made up of a product of two forms. What are these two forms?

1. linear factors

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Question

What is the condition of the discriminant for irreducible quadratics?

The discriminant is less than zero: D = b2 - 4ac < 0

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Question

For a polynomial of degree n = 5, what are the combinations of the types of roots it can have? Choose any that apply.

1. all pairs of complex conjugate roots
2. all real roots
3. 3 real roots and 2 pairs of complex conjugate roots
4. 5 real roots and a pair of complex conjugate roots.

Here, n = 7 is an odd number so we can have ii, iii and iv.

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Question

For a polynomial of degree n = 4, what are the combinations of the types of roots it can have? Choose any that apply.

1. all pairs of complex conjugate roots
2. all real roots
3. 2 real roots and a pair of complex conjugate roots
4. all of the above

As n = 4 is even, we obtain iv. all of the above.

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Question

What are the two main applications of the Fundamental Theorem of Algebra?

1. identifying the number of roots in a polynomial
2. finding the zeros snd degree of a polynomial

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Question

An irreducible quadratic is one that we can no longer break down into a product of linear equations.

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Question

If the discriminant is more than zero, D = b2 - 4ac > 0, what does this mean for a quadratic polynomial p(x) and what type of roots do we obtain?

The polynomial p(x) can be reduced further into a product of linear factors. We will obtain two distinct real roots.

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Question

Using the Fundamental Theorem of Algebra, how many roots would we have for the polynomial f(x) = 2x9 + x + 4?

Since the degree of f(x) is 9, we will obtain 9 roots.

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Question

Using the Fundamental Theorem of Algebra, how many roots would we have for the polynomial f(x) = 3x2 + 2x5 - 7x - 5x3 + 6?

Since the degree of f(x) is 5, we will gain 5 roots.

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Question

What is the name of the concept used to find all possible rational zeros of a polynomial?

The Rational Zeros Theorem

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Question

What is a rational zero?

A rational zero is a rational number written as a fraction of two integers

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Question

What can the Rational Zeros Theorem tell us about a polynomial?

The Rational Zeros Theorem can help us find all possible rational zeros of a given polynomial

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Question

What does the variable p represent in the Rational Zeros Theorem?

The numerator p represents a factor of the constant term in a given polynomial

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Question

What does the variable q represent in the Rational Zeros Theorem?

The denominator q represents a factor of the leading coefficient in a given polynomial

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Question

Before applying the Rational Zeros Theorem to a given polynomial, what is an important step to first consider? Choose one of the following choices.

1. conduct the grouping method
2. use trial and error
3. factor out the greatest common divisor
4. rearrange the variables in descending order of degree

C. factor out the greatest common divisor

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Question

Does the Rational Zeros Theorem give us the correct set of solutions that satisfy a given polynomial?

No. The Rational Zeros Theorem only provides all possible rational roots of a given polynomial.

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Question

How do you correctly determine the set of rational zeros that satisfy the given polynomial after applying the Rational Zeros Theorem?

Conduct synthetic division to calculate the polynomial at each value of rational zeros found.

Repeat this process until a quadratic quotient is reached or can be factored easily.

Set all factors equal to zero and solve to find the remaining solutions.

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Question

What is an irrational zero?

An irrational zero is a number that is not rational and is represented by an infinitely non-repeating decimal

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Question

Why is it important to use the Rational Zeros Theorem to find rational zeros of a given polynomial?

Once we have found the rational zeros, we can easily factorize and solve polynomials by recognizing the solutions of a given polynomial

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Question

What is a cubic function? State the general form of a cubic equation.

A cubic function is a polynomial of degree three. The general form of a cubic equation is $$ax^3+bx^2+cx+d=0$$ where $$a \ne 0$$.

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Question

True or false: Cubic functions have axes of symmetry.

False.

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Question

What are the first two steps when plotting cubic functions?

Find the $$x$$ and $$y$$ intercepts.

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Question

For the cubic function $$y=ax^3$$, what does varying the coefficient $$a$$ do to the graph of the cubic?

Varying the coefficient $$a$$ can invert the graph, make it steeper, or make it flatter.

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Question

For the cubic function $$y=x^3+k$$, what does varying the coefficient $$k$$ do to the graph of the cubic?

Moves the graph up or down.

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Question

For the cubic function $$y=(x-h)^3$$, what does varying the coefficient $$h$$ do to the graph of the cubic?

Shifts the graph left or right.

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Question

For the cubic function $$y=ax^3$$ when $$a$$ is large and positive, how does the cubic graph change?

The cubic graph becomes steeper.

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Question

For the cubic function $$y=ax^3$$, when $$a$$ is small and positive, how does the cubic graph change?

The cubic graph becomes flatter.

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Question

For the cubic function $$y=ax^3$$, when $$a$$ is negative, how does the cubic graph change?

The cubic graph becomes inverted.

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Question

For the cubic function $$y=x^3+k$$, when $$k$$ is positive, how does the cubic graph shift?

The cubic graph moves up the $$y$$-axis

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Question

For the cubic function $$y=x^3+k$$, when $$k$$ is negative, how does the cubic graph shift?

The cubic graph moves down on the $$y$$-axis

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Question

For the cubic function $$y=(x-h)^3$$, when $$h$$ is positive, how does the cubic graph shift?

The cubic graph moves to the right on the $$x$$-axis

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Question

For the cubic function $$y = (x-h)^3$$, when $$h$$ is negative, how does the cubic graph shift?

The cubic graph moves to the left on the $$x$$-axis.

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Question

How many turning points can cubic graphs have?

0 or 2.

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Question

What are the steps in plotting cubic functions?

1. Find the $$x$$-intercepts
2. Locate the $$y$$-intercepts
3. Create a table of values
4. Identify the minimum and maximum points
5. Sketch the graph

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Question

What is the Location Principle in the context of a cubic equation?

Suppose $$y = f(x)$$ is a cubic polynomial function. If $$a$$ and $$b$$ are two values such that $$f(a) < 0$$ and $$f(b) > 0$$, then the function has at least one real zero between $$a$$ and $$b$$ .

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