name:
looking at polynomials and their factors
1. factor the following polynomials:
a. $x^2 + 14x + 49 = ( )( )$
b. $x^2 + 12x + 32 = ( )( )$
c. $x^2 + 10x + 21 = ( )( )$
2. what do the three polynomials above have in common?

3. what do you notice about the signs of the numbers in the binomials?

4. factor these polynomials:
a. $x^2 - 12x + 32 = ( )( )$
b. $x^2 - 12x + 27 = ( )( )$
c. $x^2 - 11x + 24 = ( )( )$
5. what do the three polynomials above have in common?

6. what do you notice about the signs of the numbers in the binomials?

7. factor these polynomials:
a. $x^2 - 9x - 10 = ( )( )$
b. $x^2 + 8x - 33 = ( )( )$
c. $x^2 - 3x - 54 = ( )( )$
8. what do the three polynomials above have in common?

9. what do you notice about the signs of the numbers in the binomials?

Question

name:
looking at polynomials and their factors
1. factor the following polynomials:
a. $x^2 + 14x + 49 = ( )( )$
b. $x^2 + 12x + 32 = ( )( )$
c. $x^2 + 10x + 21 = ( )( )$
2. what do the three polynomials above have in common?

3. what do you notice about the signs of the numbers in the binomials?

4. factor these polynomials:
a. $x^2 - 12x + 32 = ( )( )$
b. $x^2 - 12x + 27 = ( )( )$
c. $x^2 - 11x + 24 = ( )( )$
5. what do the three polynomials above have in common?

6. what do you notice about the signs of the numbers in the binomials?

7. factor these polynomials:
a. $x^2 - 9x - 10 = ( )( )$
b. $x^2 + 8x - 33 = ( )( )$
c. $x^2 - 3x - 54 = ( )( )$
8. what do the three polynomials above have in common?

9. what do you notice about the signs of the numbers in the binomials?

Accepted Answer

### Question 1a:
# Explanation:
## Step1: Identify the form
The polynomial $x^2 + 14x + 49$ is a perfect square trinomial. The form of a perfect square trinomial is $a^2 + 2ab + b^2=(a + b)^2$. Here, $a = x$, and $2ab = 14x$, so $2b = 14$ which gives $b = 7$. And $b^2=49$.
## Step2: Factor
So $x^2 + 14x + 49=(x + 7)(x + 7)$

# Answer: $(x + 7)(x + 7)$

### Question 1b:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $32$ and add up to $12$. The numbers are $8$ and $4$ since $8\times4 = 32$ and $8 + 4=12$.
## Step2: Factor
So $x^2+12x + 32=(x + 8)(x + 4)$

# Answer: $(x + 8)(x + 4)$

### Question 1c:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $21$ and add up to $10$. The numbers are $7$ and $3$ since $7\times3=21$ and $7 + 3 = 10$.
## Step2: Factor
So $x^2+10x + 21=(x + 7)(x + 3)$

# Answer: $(x + 7)(x + 3)$

### Question 2:
# Brief Explanations:
All three polynomials are quadratic trinomials of the form $x^{2}+bx + c$ where $b$ and $c$ are positive. Also, they can be factored into two binomials of the form $(x + m)(x + n)$ where $m$ and $n$ are positive integers (in 1a, $m=n = 7$; in 1b, $m = 8,n=4$; in 1c, $m = 7,n = 3$).

# Answer: They are quadratic trinomials $x^{2}+bx + c$ with positive $b,c$, factorable into $(x + m)(x + n)$ ($m,n>0$)

### Question 3:
# Brief Explanations:
In the binomials obtained from factoring the polynomials in question 1, both the constant terms (the non - $x$ terms) in each binomial are positive. For example, in $(x + 7)(x + 7)$, $7$ is positive; in $(x + 8)(x + 4)$, $8$ and $4$ are positive; in $(x + 7)(x + 3)$, $7$ and $3$ are positive.

# Answer: The constant terms in the binomials are positive.

### Question 4a:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $32$ and add up to $- 12$ (since the coefficient of $x$ is $-12$). The numbers are $-8$ and $-4$ since $(-8)\times(-4)=32$ and $(-8)+(-4)=-12$.
## Step2: Factor
So $x^{2}-12x + 32=(x - 8)(x - 4)$

# Answer: $(x - 8)(x - 4)$

### Question 4b:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $27$ and add up to $-12$. The numbers are $-9$ and $-3$ since $(-9)\times(-3)=27$ and $(-9)+(-3)=-12$.
## Step2: Factor
So $x^{2}-12x + 27=(x - 9)(x - 3)$

# Answer: $(x - 9)(x - 3)$

### Question 4c:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $24$ and add up to $-11$. The numbers are $-8$ and $-3$ since $(-8)\times(-3)=24$ and $(-8)+(-3)=-11$.
## Step2: Factor
So $x^{2}-11x + 24=(x - 8)(x - 3)$

# Answer: $(x - 8)(x - 3)$

### Question 5:
# Brief Explanations:
All three polynomials are quadratic trinomials of the form $x^{2}-bx + c$ where $b$ is positive and $c$ is positive. They can be factored into two binomials of the form $(x - m)(x - n)$ where $m$ and $n$ are positive integers (in 4a, $m = 8,n = 4$; in 4b, $m=9,n = 3$; in 4c, $m = 8,n=3$).

# Answer: They are quadratic trinomials $x^{2}-bx + c$ ($b,c>0$), factorable into $(x - m)(x - n)$ ($m,n>0$)

### Question 6:
# Brief Explanations:
In the binomials obtained from factoring the polynomials in question 4, both the constant terms (the non - $x$ terms) in each binomial are negative? No, wait. Wait, the binomials are of the form $(x - m)(x - n)$ where $m$ and $n$ are positive. So the constant terms in the binomials (the terms without $x$) are negative? No, $(x - m)$ has a constant term of $-m$ (negative) and $(x - n)$ has a constant term of $-n$ (negative). Wait, actually, when we factor $x^{2}-bx + c$ with $b>0$ and $c>0$, we get $(x - m)(x - n)$ where $m + n=b$ and $m\times n=c$, so $m$ and $n$ are positive, and the constant terms in the binomials are negative.

# Answer: The constant terms in the binomials are negative.

### Question 7a:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $-10$ and add up to $-9$. The numbers are $-10$ and $1$ since $(-10)\times1=-10$ and $(-10)+1=-9$.
## Step2: Factor
So $x^{2}-9x - 10=(x - 10)(x + 1)$

# Answer: $(x - 10)(x + 1)$

### Question 7b:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $-33$ and add up to $8$. The numbers are $11$ and $-3$ since $11\times(-3)=-33$ and $11+(-3)=8$.
## Step2: Factor
So $x^{2}+8x - 33=(x + 11)(x - 3)$

# Answer: $(x + 11)(x - 3)$

### Question 7c:
# Explanation:
## Step1: Find two numbers
We need two numbers that multiply to $-54$ and add up to $-3$. The numbers are $-9$ and $6$ since $(-9)\times6=-54$ and $(-9)+6=-3$.
## Step2: Factor
So $x^{2}-3x - 54=(x - 9)(x + 6)$

# Answer: $(x - 9)(x + 6)$

### Question 8:
# Brief Explanations:
All three polynomials are quadratic trinomials of the form $x^{2}+bx - c$ (or $x^{2}-bx - c$) where the constant term $c$ is positive and the coefficient of $x$ ($b$) can be positive or negative. They can be factored into two binomials of the form $(x + m)(x - n)$ or $(x - m)(x + n)$ where one of $m$ or $n$ is positive and the other is negative, and $|m - n|=$ the absolute value of the coefficient of $x$, and $m\times n=$ the absolute value of the constant term.

# Answer: They are quadratic trinomials $x^{2}+bx - c$ (or $x^{2}-bx - c$) with $c>0$, factorable into binomials with one positive and one negative constant term.

### Question 9:
# Brief Explanations:
In the binomials obtained from factoring the polynomials in question 7, one of the constant terms (the non - $x$ terms) in each binomial is positive and the other is negative. For example, in $(x - 10)(x + 1)$, $-10$ is negative and $1$ is positive; in $(x + 11)(x - 3)$, $11$ is positive and $-3$ is negative; in $(x - 9)(x + 6)$, $-9$ is negative and $6$ is positive.

# Answer: One constant term in each binomial is positive and the other is negative.

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QUESTION IMAGE

Question

Explanation:

Question 1a:

Step1: Identify the form

Step2: Factor

Step1: Find two numbers

Step2: Factor

Step1: Find two numbers

Step2: Factor

Answer:

Question 1b: