ap precalculus unit 1: polynomial functions & complex zeros
name:
date: 4/18 - 25 period:
directions: for each of the following polynomials, state the degree of the polynomial, determine all real zeros of the polynomial and the multiplicity of each zero.
1. $f(x)=4(x + 3)^2(x - 1)$
degree of polynomial:
zeros (with multiplicity):
2. $g(x)=-2x(x - 5)^3(x + 1)^2$
degree of polynomial:
zeros (with multiplicity):
3. $h(x)=x^3(x + 4)(x - e)$
degree of polynomial:
zeros (with multiplicity):
4. $k(x)=-3(x + 7)^4(2x + 1)^2$
degree of polynomial:
zeros (with multiplicity):
5. $y=(x^2 - 9)(x^2 - 4x + 4)$
degree of polynomial:
zeros (with multiplicity):
6. $p(x)=x^3+10x^2 + 25x$
degree of polynomial:
zeros (with multiplicity):
7. $m(x)=(x^2 - 1)(x^2 + 2x + 1)$
degree of polynomial:
zeros (with multiplicity):
8. $j(x)=4x^3(x^2 - 2x - 8)(x^2 + x - 20)$
degree of polynomial:
zeros (with multiplicity):

Question

Accepted Answer

### 1. For $f(x)=4(x + 3)^{2}(x - 1)$
# Explanation:
## Step1: Find the degree of the polynomial
The degree of a polynomial in factored - form $a(x - r_1)^{n_1}(x - r_2)^{n_2}\cdots(x - r_k)^{n_k}$ is the sum of the exponents $n_i$. Here, $n_1 = 2$ and $n_2=1$, so the degree is $2 + 1=3$.
## Step2: Find the zeros and their multiplicities
Set $f(x)=0$. Then $(x + 3)^{2}=0$ gives $x=-3$ with multiplicity $2$, and $(x - 1)=0$ gives $x = 1$ with multiplicity $1$.
# Answer:
Degree of polynomial: $3$
Zeros (with multiplicity): $x=-3$ (multiplicity $2$), $x = 1$ (multiplicity $1$)

### 2. For $g(x)=-2x(x - 5)^{3}(x + 1)^{2}$
# Explanation:
## Step1: Find the degree of the polynomial
The exponents are $1$ for $x$, $3$ for $(x - 5)$ and $2$ for $(x + 1)$. The degree is $1+3 + 2=6$.
## Step2: Find the zeros and their multiplicities
Set $g(x)=0$. Then $x = 0$ with multiplicity $1$, $x=5$ with multiplicity $3$ and $x=-1$ with multiplicity $2$.
# Answer:
Degree of polynomial: $6$
Zeros (with multiplicity): $x = 0$ (multiplicity $1$), $x = 5$ (multiplicity $3$), $x=-1$ (multiplicity $2$)

### 3. For $h(x)=x^{3}(x + 4)(x - e)$
# Explanation:
## Step1: Find the degree of the polynomial
The exponents are $3$ for $x^{3}$, $1$ for $(x + 4)$ and $1$ for $(x - e)$. The degree is $3+1 + 1=5$.
## Step2: Find the zeros and their multiplicities
Set $h(x)=0$. Then $x = 0$ with multiplicity $3$, $x=-4$ with multiplicity $1$ and $x = e$ with multiplicity $1$.
# Answer:
Degree of polynomial: $5$
Zeros (with multiplicity): $x = 0$ (multiplicity $3$), $x=-4$ (multiplicity $1$), $x = e$ (multiplicity $1$)

### 4. For $k(x)=-3(x + 7)^{4}(2x+1)^{2}$
# Explanation:
## Step1: Find the degree of the polynomial
The exponents are $4$ for $(x + 7)$ and $2$ for $(2x + 1)$. The degree is $4+2=6$.
## Step2: Find the zeros and their multiplicities
Set $k(x)=0$. Then $x=-7$ with multiplicity $4$ and $2x + 1=0$ gives $x=-\frac{1}{2}$ with multiplicity $2$.
# Answer:
Degree of polynomial: $6$
Zeros (with multiplicity): $x=-7$ (multiplicity $4$), $x=-\frac{1}{2}$ (multiplicity $2$)

### 5. For $y=(x^{2}-9)(x^{2}-4x + 4)$
# Explanation:
## Step1: Factor the polynomials
$x^{2}-9=(x + 3)(x - 3)$ and $x^{2}-4x + 4=(x - 2)^{2}$.
## Step2: Find the degree of the polynomial
The degree of $(x + 3)(x - 3)(x - 2)^{2}$ is $1+1 + 2=4$.
## Step3: Find the zeros and their multiplicities
Set $y = 0$. Then $x=-3$ with multiplicity $1$, $x = 3$ with multiplicity $1$ and $x = 2$ with multiplicity $2$.
# Answer:
Degree of polynomial: $4$
Zeros (with multiplicity): $x=-3$ (multiplicity $1$), $x = 3$ (multiplicity $1$), $x = 2$ (multiplicity $2$)

### 6. For $p(x)=x^{3}+10x^{2}+25x$
# Explanation:
## Step1: Factor the polynomial
$p(x)=x(x^{2}+10x + 25)=x(x + 5)^{2}$.
## Step2: Find the degree of the polynomial
The degree is $1+2=3$.
## Step3: Find the zeros and their multiplicities
Set $p(x)=0$. Then $x = 0$ with multiplicity $1$ and $x=-5$ with multiplicity $2$.
# Answer:
Degree of polynomial: $3$
Zeros (with multiplicity): $x = 0$ (multiplicity $1$), $x=-5$ (multiplicity $2$)

### 7. For $m(x)=(x^{2}-1)(x^{2}+2x + 1)$
# Explanation:
## Step1: Factor the polynomials
$x^{2}-1=(x + 1)(x - 1)$ and $x^{2}+2x + 1=(x + 1)^{2}$. So $m(x)=(x + 1)^{3}(x - 1)$.
## Step2: Find the degree of the polynomial
The degree is $3+1=4$.
## Step3: Find the zeros and their multiplicities
Set $m(x)=0$. Then $x=-1$ with multiplicity $3$ and $x = 1$ with multiplicity $1$.
# Answer:
Degree of polynomial: $4$
Zeros (with multiplicity): $x=-1$ (multiplicity $3$), $x = 1$ (multiplicity $1$)

### 8. For $j(x)=4x^{3}(x^{2}-2x - 8)(x^{2}+x - 20)$
# Explanation:
## Step1: Factor the quadratic polynomials
$x^{2}-2x - 8=(x - 4)(x+2)$ and $x^{2}+x - 20=(x + 5)(x - 4)$. So $j(x)=4x^{3}(x - 4)^{2}(x + 2)(x + 5)$.
## Step2: Find the degree of the polynomial
The degree is $3+2+1 + 1=7$.
## Step3: Find the zeros and their multiplicities
Set $j(x)=0$. Then $x = 0$ with multiplicity $3$, $x = 4$ with multiplicity $2$, $x=-2$ with multiplicity $1$ and $x=-5$ with multiplicity $1$.
# Answer:
Degree of polynomial: $7$
Zeros (with multiplicity): $x = 0$ (multiplicity $3$), $x = 4$ (multiplicity $2$), $x=-2$ (multiplicity $1$), $x=-5$ (multiplicity $1$)

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QUESTION IMAGE

Question

Explanation:

1. For \(f(x)=4(x + 3)^{2}(x - 1)\)

Step1: Find the degree of the polynomial

Step2: Find the zeros and their multiplicities

Step1: Find the degree of the polynomial

Step2: Find the zeros and their multiplicities

Step1: Find the degree of the polynomial

Step2: Find the zeros and their multiplicities

Answer:

2. For \(g(x)=-2x(x - 5)^{3}(x + 1)^{2}\)