4.3 pollys problems
mrs. gallas neighbor, polly ngo-miel, is learning about multiplication and division in her 4th grade class. she is
stuck on her homework problems and needs some help. after helping polly with her homework, mrs. gallas
wonders how the multiplication and division that polly is learning in 4th grade relates to the polynomials shes
teaching in her class.
1. a. lets see how this would work with poly-nomials. find the product of $(2x + 1)(x^{2}+4x - 3)$.
b. what is $(2x^{3}+9x^{2}-2x - 3)+(2x + 1)$? how do you know?
2. a. fill in the missing blanks on the diagram.
b. what multiplication problem could we use the diagram to solve?
c. what division problem could we use the diagram to solve?
3. use the rectangle diagram to find $(3x^{3}-20x^{2}-3x + 10)+(3x - 2)$.

Question

Accepted Answer

1. a.
# Explanation:
## Step1: Use the distributive property (FOIL - First, Outer, Inner, Last for binomial - polynomial)
$$(2x + 1)(x^{2}+4x - 3)=2x(x^{2}+4x - 3)+1(x^{2}+4x - 3)$$
## Step2: Distribute each term
$$=2x\cdot x^{2}+2x\cdot4x-2x\cdot3 + 1\cdot x^{2}+1\cdot4x - 1\cdot3$$
## Step3: Simplify using the rule $a^{m}\cdot a^{n}=a^{m + n}$
$$=2x^{3}+8x^{2}-6x+x^{2}+4x - 3$$
## Step4: Combine like - terms
$$=2x^{3}+(8x^{2}+x^{2})+( - 6x + 4x)-3$$
$$=2x^{3}+9x^{2}-2x - 3$$
# Answer:
$2x^{3}+9x^{2}-2x - 3$

1. b.
# Explanation:
## Step1: Recall the relationship between multiplication and division of polynomials
If $A\times B = C$, then $C\div A=B$ (assuming $A
eq0$). From part (a), we know that $(2x + 1)(x^{2}+4x - 3)=2x^{3}+9x^{2}-2x - 3$
## Step2: Identify the quotient
So, $(2x^{3}+9x^{2}-2x - 3)\div(2x + 1)=x^{2}+4x - 3$
# Answer:
$x^{2}+4x - 3$

2. a.
# Explanation:
## Step1: For the first column (when multiplied by $x$ gives $4x^{3}$ and by $1$ gives $4x^{2}$)
Let the first term be $4x^{2}$ (since $4x^{2}\cdot x = 4x^{3}$ and $4x^{2}\cdot1 = 4x^{2}$)
## Step2: For the second column (when multiplied by $x$ gives $-5x^{2}$ and by $1$ gives $-5x$)
Let the second term be $-5x$ (since $-5x\cdot x=-5x^{2}$ and $-5x\cdot1=-5x$)
## Step3: For the third column (when multiplied by $x$ gives $3x$ and by $1$ gives $3$)
Let the third term be $3$ (since $3\cdot x = 3x$ and $3\cdot1 = 3$)
# Answer:
The terms are $4x^{2},-5x,3$

2. b.
# Explanation:
## Step1: Use the area of a rectangle formula $A=(l)(w)$
The length is $(x + 1)$ and the width is $(4x^{2}-5x + 3)$
# Answer:
$(x + 1)(4x^{2}-5x + 3)$

2. c.
# Explanation:
## Step1: Use the relationship between multiplication and division
If $A=(l)(w)$, then $A\div l=w$ or $A\div w = l$
# Answer:
$(4x^{3}-x^{2}-2x + 3)\div(x + 1)$ or $(4x^{3}-x^{2}-2x + 3)\div(4x^{2}-5x + 3)$

3.
# Explanation:
## Step1: Set up the rectangle diagram for polynomial long - division (not shown in detail here, but conceptually similar to the area model)
We know that $(3x-2)(ax^{2}+bx + c)=3x^{3}-20x^{2}-3x + 10$
First, $3x\cdot ax^{2}=3x^{3}$, so $a = 1$
Then $(3x-2)(x^{2}+bx + c)=3x^{3}+(3b)x^{2}+(3c)x-2x^{2}-2bx-2c$
$=3x^{3}+(3b - 2)x^{2}+(3c-2b)x-2c$
Equating coefficients:
$3b-2=-20$, so $3b=-18$, $b=-6$
$3c-2b=-3$, substituting $b = - 6$, we get $3c+12=-3$, $3c=-15$, $c=-5$
# Answer:
$x^{2}-6x - 5$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Use the distributive property (FOIL - First, Outer, Inner, Last for binomial - polynomial)

Step2: Distribute each term

Step3: Simplify using the rule \(a^{m}\cdot a^{n}=a^{m + n}\)

Step4: Combine like - terms

Step1: Recall the relationship between multiplication and division of polynomials

Step2: Identify the quotient

Step1: For the first column (when multiplied by \(x\) gives \(4x^{3}\) and by \(1\) gives \(4x^{2}\))

Step2: For the second column (when multiplied by \(x\) gives \(-5x^{2}\) and by \(1\) gives \(-5x\))

Step3: For the third column (when multiplied by \(x\) gives \(3x\) and by \(1\) gives \(3\))

Answer: