name
date:
period:
multiplying polynomials practice 2
directions: multiply the polynomials. write your answer in standard form.
1. $(-x^3)(8x^5 + xy + 4y)$
2. $(x + 7)(x - 3)$
3. $(2x - 2)(4x + 1)$
4. $5x^3(2x^6 - xy^4 + 2y^6)$
5. $-7y^5(3y^2 + y + 2)$
6. $(3x - 4y)(6x + 8y)$
7. $(x + 10)^2$
8. $(9x^2y^6)(x^3 - xy^4 + 2y^7)$
9. find the expression that represents the area of the triangle.
(2x - 4)
(x + 8)

Question

Accepted Answer

### Problem 1: $(-x^3)(8x^5 + xy + 4y)$
# Explanation:
## Step1: Distribute $-x^3$
Multiply $-x^3$ with each term inside the parentheses:
$-x^3 \cdot 8x^5 = -8x^{3+5} = -8x^8$
$-x^3 \cdot xy = -x^{3+1}y = -x^4y$
$-x^3 \cdot 4y = -4x^3y$

## Step2: Combine terms
Combine the results to get the standard form:
$-8x^8 - x^4y - 4x^3y$

# Answer: $-8x^8 - x^4y - 4x^3y$

### Problem 2: $(x + 7)(x - 3)$
# Explanation:
## Step1: Use FOIL method
Multiply First, Outer, Inner, Last terms:
First: $x \cdot x = x^2$
Outer: $x \cdot (-3) = -3x$
Inner: $7 \cdot x = 7x$
Last: $7 \cdot (-3) = -21$

## Step2: Combine like terms
Combine $-3x$ and $7x$:
$x^2 + (-3x + 7x) - 21 = x^2 + 4x - 21$

# Answer: $x^2 + 4x - 21$

### Problem 3: $(2x - 2)(4x + 1)$
# Explanation:
## Step1: Distribute each term
Multiply $2x$ and $-2$ with $4x$ and $1$:
$2x \cdot 4x = 8x^2$
$2x \cdot 1 = 2x$
$-2 \cdot 4x = -8x$
$-2 \cdot 1 = -2$

## Step2: Combine like terms
Combine $2x$ and $-8x$:
$8x^2 + (2x - 8x) - 2 = 8x^2 - 6x - 2$

# Answer: $8x^2 - 6x - 2$

### Problem 4: $5x^3(2x^6 - xy^4 + 2y^6)$
# Explanation:
## Step1: Distribute $5x^3$
Multiply $5x^3$ with each term:
$5x^3 \cdot 2x^6 = 10x^{3+6} = 10x^9$
$5x^3 \cdot (-xy^4) = -5x^{3+1}y^4 = -5x^4y^4$
$5x^3 \cdot 2y^6 = 10x^3y^6$

## Step2: Combine terms
Result in standard form:
$10x^9 - 5x^4y^4 + 10x^3y^6$

# Answer: $10x^9 - 5x^4y^4 + 10x^3y^6$

### Problem 5: $-7y^5(3y^2 + y + 2)$
# Explanation:
## Step1: Distribute $-7y^5$
Multiply with each term:
$-7y^5 \cdot 3y^2 = -21y^{5+2} = -21y^7$
$-7y^5 \cdot y = -7y^{5+1} = -7y^6$
$-7y^5 \cdot 2 = -14y^5$

## Step2: Combine terms
Standard form:
$-21y^7 - 7y^6 - 14y^5$

# Answer: $-21y^7 - 7y^6 - 14y^5$

### Problem 6: $(3x - 4y)(6x + 8y)$
# Explanation:
## Step1: Distribute each term
Multiply $3x$ and $-4y$ with $6x$ and $8y$:
$3x \cdot 6x = 18x^2$
$3x \cdot 8y = 24xy$
$-4y \cdot 6x = -24xy$
$-4y \cdot 8y = -32y^2$

## Step2: Combine like terms
Combine $24xy$ and $-24xy$:
$18x^2 + (24xy - 24xy) - 32y^2 = 18x^2 - 32y^2$

# Answer: $18x^2 - 32y^2$

### Problem 7: $(x + 10)^2$
# Explanation:
## Step1: Expand using $(a + b)^2 = a^2 + 2ab + b^2$
Here, $a = x$, $b = 10$:
$x^2 + 2 \cdot x \cdot 10 + 10^2 = x^2 + 20x + 100$

# Answer: $x^2 + 20x + 100$

### Problem 8: $(9x^2y^6)(x^3 - xy^4 + 2y^7)$
# Explanation:
## Step1: Distribute $9x^2y^6$
Multiply with each term:
$9x^2y^6 \cdot x^3 = 9x^{2+3}y^6 = 9x^5y^6$
$9x^2y^6 \cdot (-xy^4) = -9x^{2+1}y^{6+4} = -9x^3y^{10}$
$9x^2y^6 \cdot 2y^7 = 18x^2y^{6+7} = 18x^2y^{13}$

## Step2: Combine terms
Standard form:
$9x^5y^6 - 9x^3y^{10} + 18x^2y^{13}$

# Answer: $9x^5y^6 - 9x^3y^{10} + 18x^2y^{13}$

### Problem 9: Area of Triangle with legs $(2x - 4)$ and $(x + 8)$
# Explanation:
## Step1: Recall triangle area formula
Area of a right triangle: $A = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{height}$

## Step2: Substitute base and height
Base = $x + 8$, Height = $2x - 4$:
$A = \frac{1}{2}(x + 8)(2x - 4)$

## Step3: Simplify $(x + 8)(2x - 4)$
Distribute:
$x \cdot 2x + x \cdot (-4) + 8 \cdot 2x + 8 \cdot (-4) = 2x^2 - 4x + 16x - 32 = 2x^2 + 12x - 32$

## Step4: Multiply by $\frac{1}{2}$
$\frac{1}{2}(2x^2 + 12x - 32) = x^2 + 6x - 16$

# Answer: $x^2 + 6x - 16$

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QUESTION IMAGE

Question

Explanation:

Problem 1: $(-x^3)(8x^5 + xy + 4y)$

Step1: Distribute $-x^3$

Step2: Combine terms

Step1: Use FOIL method

Step2: Combine like terms

Step1: Distribute each term

Step2: Combine like terms

Answer:

Problem 2: $(x + 7)(x - 3)$