1 identify and factor the difference of two perfect squares.
2 look at the expression below.
-4x² + 16x - 24 + 6x
which of the following cannot be a pair of factors of the expression?
a. -2, x - 3
b. -2, 2x - 3
c. 2, 4 - x
d. 2, 2x - 3
3 michelle factored x² - 6x + 9 as follows:
x² - 6x + 9 =
x² - 2(x·3) + 3² =
(x - 3)(x + 3)
the correct factorization should be:
a. (x - 3)(x + 3)
b. (x + 3)²
c. (x - 3)²
4 which of these statements accurately describes the graph of the function f(x) = x² + 20x + 100 ?
a. the graph passes through the x - axis at -10.
b. the graph passes through the x - axis at 10.
c. the graph passes through the y - axis at -10.
d. the graph passes through the y - axis at 10.
5 simplify:
(2x - 10)/(10x - 2)
a. (x - 5)/(5x - 1)
b. (1/5)x + 5
c. -1
d. -4⁴/₅
6 simplify.
(3x² + 14x + 8)/(x + 4)
a. 3x² + 13x + 4
b. 3x + 2
c. 1/(3x + 2)
d. 3x + 4
7 for which values of x is the following expression undefined?
(x + 2)/(x² - 25)
a. -2 only
b. 5 only
c. 5 and -5
d. 5, -5, and -2

Question

1 identify and factor the difference of two perfect squares.
2 look at the expression below.
  -4x² + 16x - 24 + 6x
which of the following cannot be a pair of factors of the expression?
a. -2, x - 3
b. -2, 2x - 3
c. 2, 4 - x
d. 2, 2x - 3
3 michelle factored x² - 6x + 9 as follows:
x² - 6x + 9 =
x² - 2(x·3) + 3² =
(x - 3)(x + 3)
the correct factorization should be:
a. (x - 3)(x + 3)
b. (x + 3)²
c. (x - 3)²
4 which of these statements accurately describes the graph of the function f(x) = x² + 20x + 100 ?
a. the graph passes through the x - axis at -10.
b. the graph passes through the x - axis at 10.
c. the graph passes through the y - axis at -10.
d. the graph passes through the y - axis at 10.
5 simplify:
(2x - 10)/(10x - 2)
a. (x - 5)/(5x - 1)
b. (1/5)x + 5
c. -1
d. -4⁴/₅
6 simplify.
(3x² + 14x + 8)/(x + 4)
a. 3x² + 13x + 4
b. 3x + 2
c. 1/(3x + 2)
d. 3x + 4
7 for which values of x is the following expression undefined?
(x + 2)/(x² - 25)
a. -2 only
b. 5 only
c. 5 and -5
d. 5, -5, and -2

Accepted Answer

### Question 5
# Explanation:
## Step1: Factor numerator and denominator
Factor out 2 from numerator: $2x - 10 = 2(x - 5)$
Factor out 2 from denominator: $10x - 2 = 2(5x - 1)$
## Step2: Simplify the fraction
Cancel out the common factor 2: $\frac{2(x - 5)}{2(5x - 1)} = \frac{x - 5}{5x - 1}$
# Answer: A. $\frac{x - 5}{5x - 1}$

### Question 6
# Explanation:
## Step1: Factor the numerator
Factor $3x^2 + 14x + 8$. Find two numbers that multiply to $3\times8 = 24$ and add to 14. The numbers are 12 and 2.
Rewrite the middle term: $3x^2 + 12x + 2x + 8$
Group and factor: $3x(x + 4) + 2(x + 4) = (3x + 2)(x + 4)$
## Step2: Simplify the fraction
Cancel out the common factor $x + 4$: $\frac{(3x + 2)(x + 4)}{x + 4} = 3x + 2$
# Answer: B. $3x + 2$

### Question 7
# Explanation:
## Step1: Identify when the expression is undefined
A rational expression is undefined when the denominator is zero. So, set $x^2 - 25 = 0$
## Step2: Solve the equation
Factor the denominator: $x^2 - 25 = (x - 5)(x + 5) = 0$
Set each factor to zero: $x - 5 = 0$ or $x + 5 = 0$
Solve for $x$: $x = 5$ or $x = -5$
# Answer: C. 5 and -5

### Question 4
# Explanation:
## Step1: Factor the quadratic function
Factor $f(x) = x^2 + 20x + 100$. It is a perfect square trinomial: $x^2 + 20x + 100 = (x + 10)^2$
## Step2: Analyze the x - intercept
The x - intercept occurs when $f(x) = 0$. So, $(x + 10)^2 = 0$ gives $x = - 10$ (a repeated root). So the graph touches the x - axis at $x=-10$ (passes through in the sense of a root, even though it's a repeated root).
For the y - intercept, set $x = 0$: $f(0)=0^2 + 20(0)+100 = 100$, so the y - intercept is 100, not - 10 or 10.
So option A says the graph passes through the x - axis at - 10, which is correct.
# Answer: A. The graph passes through the x - axis at - 10.

### Question 3
# Explanation:
## Step1: Recall the perfect square trinomial formula
The perfect square trinomial formula is $a^2 - 2ab + b^2=(a - b)^2$
For $x^2 - 6x + 9$, $a = x$ and $b = 3$ (since $2ab = 2(x)(3)=6x$)
## Step2: Factor the trinomial
Using the formula, $x^2 - 6x + 9=(x - 3)^2$
Michelle factored it as $(x - 3)(x + 3)$ which is incorrect (that's the difference of squares). The correct factorization is $(x - 3)^2$
# Answer: C. $(x - 3)^2$

### Question 2
# Explanation:
## Step1: Simplify the original expression
First, combine like terms in $-4x^2 + 16x - 24 + 6x$: $-4x^2+(16x + 6x)-24=-4x^2 + 22x - 24$
Factor out -2: $-2(2x^2 - 11x + 12)$ (or factor out other common factors to check each option)
Let's check each option by multiplying the factors:
- Option A: $-2(x - 3)=-2x + 6$. If we divide $-4x^2 + 22x - 24$ by $-2$, we get $2x^2 - 11x + 12$. If we divide by $(x - 3)$, using polynomial division or factoring, $2x^2 - 11x + 12=(2x - 3)(x - 4)$? Wait, no, let's check option B: $-2(2x - 3)=-4x + 6$. Multiply $-2$ and $2x - 3$: $-4x + 6$. Now, if we divide $-4x^2 + 22x - 24$ by $-4x + 6$, let's see: $-4x^2 + 22x - 24=(-2)(2x^2 - 11x + 12)$. Let's try to factor $2x^2 - 11x + 12$. Discriminant: $121 - 96 = 25$, roots: $\frac{11\pm5}{4}$, so $\frac{16}{4}=4$ and $\frac{6}{4}=\frac{3}{2}$. So $2x^2 - 11x + 12=(2x - 3)(x - 4)$. Wait, maybe a better approach: check which pair does not multiply to the original expression.
Option D: $2(2x - 3)=4x - 6$. Multiply 2 and $2x - 3$ gives $4x - 6$. Now, if we multiply $2$ and $2x - 3$, does it factor into the original? Wait, original after combining like terms: $-4x^2 + 22x - 24=-2(2x^2 - 11x + 12)$. Let's check option C: $2(4 - x)=8 - 2x$. Multiply 2 and $4 - x$ gives $8 - 2x$. If we divide $-4x^2 + 22x - 24$ by $2$, we get $-2x^2 + 11x - 12$. If we divide by $(4 - x)$, let's do polynomial division: $-2x^2 + 11x - 12$ divided by $(4 - x)$. Let $y = 4 - x$, so $x = 4 - y$. Substitute: $-2(4 - y)^2 + 11(4 - y)-12=-2(16 - 8y + y^2)+44 - 11y - 12=-32 + 16y - 2y^2 + 32 - 11y=-2y^2 + 5y$, which is not a polynomial in $x$ nicely. Wait, maybe the error is in option B. Wait, original expression: $-4x^2 + 16x - 24 + 6x=-4x^2 + 22x - 24$. Let's factor out -2: $-2(2x^2 - 11x + 12)$. Now, factor $2x^2 - 11x + 12$: as above, discriminant 25, roots 4 and 3/2. So $2x^2 - 11x + 12=(2x - 3)(x - 4)$. So the factors of the original expression are $-2$, $2x - 3$, and $x - 4$. Now, check option B: $-2$ and $2x - 3$ are factors (since $-2(2x - 3)=-4x + 6$, and when we multiply $-4x + 6$ with $(x - 4)$, we get $-4x^2 + 16x + 6x - 24=-4x^2 + 22x - 24$, which is correct). Wait, maybe I made a mistake. Wait, the question is which CANNOT be a pair of factors. Let's check option D: $2$ and $2x - 3$. Multiply them: $2(2x - 3)=4x - 6$. Now, divide the original expression $-4x^2 + 22x - 24$ by $4x - 6$. Using polynomial long division:
Divide $-4x^2 + 22x - 24$ by $4x - 6$.
First term: $-4x^2\div4x=-x$
Multiply $-x$ by $4x - 6$: $-4x^2 + 6x$
Subtract from dividend: $(-4x^2 + 22x - 24)-(-4x^2 + 6x)=16x - 24$
Next term: $16x\div4x = 4$
Multiply 4 by $4x - 6$: $16x - 24$
Subtract: $0$. So it divides evenly. Wait, maybe option C: $2$ and $4 - x$. Multiply them: $2(4 - x)=8 - 2x$. Divide $-4x^2 + 22x - 24$ by $8 - 2x$ (which is $-2(x - 4)$). Let's factor out -2 from denominator: $-2(x - 4)$. Dividend: $-4x^2 + 22x - 24=-2(2x^2 - 11x + 12)$. So we have $\frac{-2(2x^2 - 11x + 12)}{-2(x - 4)}=\frac{2x^2 - 11x + 12}{x - 4}$. As we saw earlier, $2x^2 - 11x + 12=(2x - 3)(x - 4)$, so $\frac{(2x - 3)(x - 4)}{x - 4}=2x - 3$, which is a polynomial. Wait, maybe the answer is option B? No, earlier calculation shows it divides. Wait, maybe I messed up the original expression. The original expression is $-4x^2 + 16x - 24 + 6x$. Let's combine like terms again: $16x + 6x = 22x$, so $-4x^2 + 22x - 24$. Let's check option A: $-2$ and $x - 3$. Multiply them: $-2(x - 3)=-2x + 6$. Divide $-4x^2 + 22x - 24$ by $-2x + 6$ (which is $-2(x - 3)$). Dividend: $-4x^2 + 22x - 24$, divisor: $-2x + 6$.
First term: $-4x^2\div(-2x)=2x$
Multiply $2x$ by $-2x + 6$: $-4x^2 + 12x$
Subtract: $(-4x^2 + 22x - 24)-(-4x^2 + 12x)=10x - 24$
Next term: $10x\div(-2x)=-5$
Multiply $-5$ by $-2x + 6$: $10x - 30$
Subtract: $(10x - 24)-(10x - 30)=6$. There is a remainder of 6, so $-2$ and $x - 3$ are not factors? Wait, this contradicts earlier. Wait, maybe I made a mistake in factoring. Let's use the factor theorem. For a factor $x - a$, $f(a)=0$. Let's take option A: factor $-2$ and $x - 3$. Let's check if $x = 3$ is a root of $-4x^2 + 22x - 24$. Plug in $x = 3$: $-4(9)+22(3)-24=-36 + 66 - 24=6
eq0$, so $x - 3$ is not a factor. Wait, so option A cannot be a pair? But earlier when I multiplied $-2$ and $x - 3$, I thought it might be, but the factor theorem shows it's not. Wait, this is confusing. Wait, the original expression: $-4x^2 + 22x - 24$. Let's factor out -2: $-2(2x^2 - 11x + 12)$. Now, factor $2x^2 - 11x + 12$: as before, roots at $x = 4$ and $x = 3/2$. So $2x^2 - 11x + 12=(x - 4)(2x - 3)$. So the original expression is $-2(x - 4)(2x - 3)$. So the factors are $-2$, $x - 4$, $2x - 3$, or $2$, $-(x - 4)$, $2x - 3$ (i.e., $2$, $4 - x$, $2x - 3$), or $-2$, $x - 4$, $2x - 3$, or $2$, $x - 4$, $-(2x - 3)$ (i.e., $2$, $x - 4$, $-2x + 3$). Now, let's check each option:
- Option A: $-2$ and $x - 3$. $x - 3$ is not a factor (since root at $x = 3$ gives non - zero), so this pair cannot be factors.
- Option B: $-2$ and $2x - 3$. $2x - 3$ is a factor, and $-2$ is a factor, so this pair can be factors.
- Option C: $2$ and $4 - x$ (which is $-(x - 4)$). $4 - x$ is a factor (since $x - 4$ is a factor), and $2$ is a factor (since we can factor out 2 from $-2$ to get 2 and $-1$), so this pair can be factors.
- Option D: $2$ and $2x - 3$. $2x - 3$ is a factor, and $2$ is a factor (from factoring out -2 as 2 and -1), so this pair can be factors.

So the answer is option A.
# Answer: A. $-2$, $x - 3$

### Question 1 (General Explanation for factoring difference of two perfect squares)
# Explanation:
## Step1: Recall the formula
The difference of two perfect squares formula is $a^2 - b^2=(a - b)(a + b)$
## Step2: Identify $a$ and $b$
For a given expression in the form of difference of two perfect squares (e.g., $x^2 - 9$, where $a = x$ and $b = 3$), apply the formula.
## Step3: Factor the expression
Using the formula, $x^2 - 9=(x - 3)(x + 3)$
(Note: Since no specific expression was given in the question, this is a general method for factoring difference of two perfect squares)
# Answer: (General method: Use $a^2 - b^2=(a - b)(a + b)$ to factor)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 5

Step1: Factor numerator and denominator

Step2: Simplify the fraction

Step1: Factor the numerator

Step2: Simplify the fraction

Step1: Identify when the expression is undefined

Step2: Solve the equation

Answer:

Question 6