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Precalculus for Team-Based Inquiry Learning: 2024 Development Edition — Instructor Version

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Section 4.4 Zeros of Polynomial Functions (PR4)

Subsection 4.4.1 Activities

Remark 4.4.1.

Recall that to find the \(x\)-intercepts of a function \(f(x)\text{,}\) we need to find the values of \(x\) that make \(f(x)=0\text{.}\) We saw in Section 1.5 that the zero product property (Definition 1.5.3) was helpful when \(f(x)\) is a polynomial that we can factor.

Activity 4.4.2.

Consider the following polynomials.
(a)
What are the \(x\)-intercepts of the function \(f(x)=(x+5)(x-3)\text{?}\)
Answer.
\(-5\) and \(3\)
(b)
What are the \(x\)-intercepts of the function \(g(x)=(x+5)(x-3)(x-7)\text{?}\)
Answer.
\(-5\text{,}\) \(3\text{,}\) and \(7\)
(c)
What are the \(x\)-intercepts of the function \(h(x)=x^3-x^2-56x\text{?}\)
Hint.
\(h(x)=x(x^2-x-56)\text{.}\)
Answer.
\(-7\text{,}\) \(0\text{,}\) and \(8\)

Definition 4.4.3.

Real zeros of a polynomial function are the same as the \(x\)-intercepts.

Activity 4.4.5.

Use Theorem 4.4.4 to find the zeros of the polynomial \(f(x)=x^4-x^3\text{.}\)
Answer.
\(0\) and \(1\text{.}\)

Activity 4.4.6.

Consider the polynomial \(f(x)=2x^3-7x^2-33x+108\text{.}\)
(a)
It turns out that \(x-3\) is a factor of \(f(x)\text{.}\) Use this fact to find a polynomial \(g(x)\) such that \(f(x)=(x-3)g(x)\text{.}\)
  1. \(\displaystyle g(x)=2x^2-x-36\)
  2. \(\displaystyle g(x)=2x^2+x-36\)
  3. \(\displaystyle g(x)=2x^2-15x-27\)
  4. \(\displaystyle g(x)=2x^2+15x-27\)
Answer.
A.
(b)
Since \(f(x)=(x-3)(2x^2-x-36)\text{,}\) how can we rewrite \(f(x)\text{?.}\)
  1. \(\displaystyle f(x)=(x-3)(x+4)(2x-9)\)
  2. \(\displaystyle f(x)=(x-3)(x-4)(2x+9)\)
  3. \(\displaystyle f(x)=(x-3)(2x+3)(x-9)\)
  4. \(\displaystyle f(x)=(x-3)(2x-3)(x+9)\)
Answer.
A.
(c)
How many zeros does \(f(x)\) have?
  1. \(\displaystyle 1\)
  2. \(\displaystyle 2\)
  3. \(\displaystyle 3\)
  4. \(\displaystyle 4\)
Answer.
\(3\)

Definition 4.4.7.

The degree of a polynomial function is the highest power of \(x\) in the expanded form of the function.

Activity 4.4.8.

Consider the function \(f(x)=5x-4\text{.}\)
(a)
What is the degree of \(f(x)\text{?}\)
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 5\)
Answer.
\(1\text{.}\)
(b)
Find the zeros of \(f(x)\text{.}\) How many are there?
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 5\)
Answer.
\(1\text{.}\)

Activity 4.4.9.

Consider the function \(g(x)=5x^2+7x-6\text{.}\)
(a)
What is the degree of \(g(x)\text{?}\)
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 5\)
Answer.
\(2\text{.}\)
(b)
Find the zeros of \(g(x)\text{.}\) How many are there?
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 5\)
Answer.
\(2\text{.}\)

Activity 4.4.10.

Consider the function \(h(x)=x^5+x^4\text{.}\)
(a)
What is the degree of \(g(x)\text{?}\)
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 5\)
Answer.
\(5\text{.}\)
(b)
Find the zeros of \(h(x)\text{.}\) How many are there?
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 5\)
Answer.
There are \(2\) distinct real zeros.

Definition 4.4.11.

The multiplicity of a zero is the number of times the corresponding linear factor appears in the factored form of the polynomial function.

Observation 4.4.12.

In the polynomial \(h(x)=x^5+x^4=x^4(x+1)\text{,}\) \(0\) was a zero with multiplicity \(4\text{.}\)

Activity 4.4.13.

How many zeros does the polynomial \(f(x)=x^2+4\) have?
  1. \(\displaystyle 0\)
  2. \(\displaystyle 1\)
  3. \(\displaystyle 2\)
  4. \(\displaystyle 3\)
Answer.
There are no real zeros, but two distinct complex zeros.

Activity 4.4.15.

Consider the polynomial \(f(x)=(3x-2)(x+1)^2(x-6)^3\text{.}\)
(a)
Find all of the zeros of \(f(x)\) with their corresponding multiplicities.
Answer.
The zeros are \(\frac{2}{3}\) with multiplicty 1, \(-1\) with multiplicity 2, and \(6\) with multiplicity 3.
(b)
Find the degree of \(f(x)\text{.}\)
  1. \(\displaystyle 1\)
  2. \(\displaystyle 2\)
  3. \(\displaystyle 3\)
  4. \(\displaystyle 6\)
Answer.
D
(c)
How do all of the exponents on the factors relate to the degree of \(f(x)\text{?}\)
Answer.
The degree is the sum of the exponents.
(d)
How do the multiplicities of the zeros relate to the degree of \(f(x)\text{?}\)

Activity 4.4.16.

Find a polynomial that satisfies the following properties:
  • \(-1\) is a zero with multiplicity \(2\)
  • \(4\) is a zero with multiplicity \(1\)
  • \(7\) is a zero with multiplicity \(3\)
Answer.
One such polynomial is \((x+1)^2(x-4)(x-7)^3\text{.}\)

Activity 4.4.17.

Find the (complex) zeros for the function, \(f(x)=x^2+16\text{.}\)
Answer.
The zeros are \(4i\) and \(-4i\text{.}\)

Activity 4.4.19.

Consider the following information about a polynomial \(f(x)\text{:}\)
  • \(x=2\) is a zero with multiplicity \(1\)
  • \(x=-1\) is a zero with multiplicity \(2\)
  • \(x=i\) is a zero with multiplicity \(1\)
(a)
What is the smallest possible degree of such a polynomial \(f(x)\) with real coefficients?
  1. \(\displaystyle 2\)
  2. \(\displaystyle 3\)
  3. \(\displaystyle 4\)
  4. \(\displaystyle 5\)
  5. \(\displaystyle 6\)
Answer.
\(5\)
(b)
Write an express for such a polynomial \(f(x)\) with real coefficients of smallest possible degree.
Answer.
\(f(x)=(x-2)(x-1)^2(x^2+1)=x^5-4x^4+6x^3-6x^2+5x-2\) is one such polynomial.

Activity 4.4.20.

Consider the function \(f(x)=x^4+x^3+2x^2+4x-8\text{.}\)
(a)
Use a graphing utility to graph \(f(x)\text{.}\)
Answer.
(b)
Find all the zeros of \(f(x)\) and their corresponding multiplicities.
Answer.
\(f(x)\) has zeros at \(-2\text{,}\) \(1\text{,}\) \(-2i\text{,}\) and \(2i\text{,}\) all of multiplicity 1.

Exercises 4.4.2 Exercises

Subsection 4.4.3 Videos

It would be great to include videos down here, like in the Calculus book!
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