multiplying polynomials
common core algebra ii homework
fluency
1. multiply the following binomials and express each product as an equivalent trinomial. use an area model to help find your product, if necessary.
(a) ((x + 5)(x + 8))
(b) ((3x + 2)(2x - 7))
(c) ((5x - 2)(2x - 3))
(d) ((x^2 - 4)(x^2 + 10))
(e) ((2x^3 + 1)(5x^3 + 4))
(f) ((x^2 - 1)(x^2 - 9))

2. find each of the following products in equivalent form. use an array model to help find your final answers if you find it helpful.
(a) ((x + 5)(x^2 + 3x + 2))
(b) ((2x - 3)(4x^2 + 5x - 7))
(c) ((2x + 5)^3)

Question

multiplying polynomials
common core algebra ii homework
fluency
1. multiply the following binomials and express each product as an equivalent trinomial. use an area model to help find your product, if necessary.
(a) ((x + 5)(x + 8))
(b) ((3x + 2)(2x - 7))
(c) ((5x - 2)(2x - 3))
(d) ((x^2 - 4)(x^2 + 10))
(e) ((2x^3 + 1)(5x^3 + 4))
(f) ((x^2 - 1)(x^2 - 9))

2. find each of the following products in equivalent form. use an array model to help find your final answers if you find it helpful.
(a) ((x + 5)(x^2 + 3x + 2))
(b) ((2x - 3)(4x^2 + 5x - 7))
(c) ((2x + 5)^3)

Accepted Answer

### Problem 1(a): $(x + 5)(x + 8)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
Multiply the First terms: $x \cdot x = x^2$
Multiply the Outer terms: $x \cdot 8 = 8x$
Multiply the Inner terms: $5 \cdot x = 5x$
Multiply the Last terms: $5 \cdot 8 = 40$

## Step 2: Combine like terms
$x^2 + 8x + 5x + 40 = x^2 + 13x + 40$

# Answer:
$x^2 + 13x + 40$

### Problem 1(b): $(3x + 2)(2x - 7)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
First: $3x \cdot 2x = 6x^2$
Outer: $3x \cdot (-7) = -21x$
Inner: $2 \cdot 2x = 4x$
Last: $2 \cdot (-7) = -14$

## Step 2: Combine like terms
$6x^2 - 21x + 4x - 14 = 6x^2 - 17x - 14$

# Answer:
$6x^2 - 17x - 14$

### Problem 1(c): $(5x - 2)(2x - 3)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
First: $5x \cdot 2x = 10x^2$
Outer: $5x \cdot (-3) = -15x$
Inner: $-2 \cdot 2x = -4x$
Last: $-2 \cdot (-3) = 6$

## Step 2: Combine like terms
$10x^2 - 15x - 4x + 6 = 10x^2 - 19x + 6$

# Answer:
$10x^2 - 19x + 6$

### Problem 1(d): $(x^2 - 4)(x^2 + 10)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
First: $x^2 \cdot x^2 = x^4$
Outer: $x^2 \cdot 10 = 10x^2$
Inner: $-4 \cdot x^2 = -4x^2$
Last: $-4 \cdot 10 = -40$

## Step 2: Combine like terms
$x^4 + 10x^2 - 4x^2 - 40 = x^4 + 6x^2 - 40$

# Answer:
$x^4 + 6x^2 - 40$

### Problem 1(e): $(2x^3 + 1)(5x^3 + 4)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
First: $2x^3 \cdot 5x^3 = 10x^6$
Outer: $2x^3 \cdot 4 = 8x^3$
Inner: $1 \cdot 5x^3 = 5x^3$
Last: $1 \cdot 4 = 4$

## Step 2: Combine like terms
$10x^6 + 8x^3 + 5x^3 + 4 = 10x^6 + 13x^3 + 4$

# Answer:
$10x^6 + 13x^3 + 4$

### Problem 1(f): $(x^2 - 1)(x^2 - 9)$
# Explanation:
## Step 1: Apply the distributive property (FOIL method)
First: $x^2 \cdot x^2 = x^4$
Outer: $x^2 \cdot (-9) = -9x^2$
Inner: $-1 \cdot x^2 = -x^2$
Last: $-1 \cdot (-9) = 9$

## Step 2: Combine like terms
$x^4 - 9x^2 - x^2 + 9 = x^4 - 10x^2 + 9$

# Answer:
$x^4 - 10x^2 + 9$

### Problem 2(a): $(x + 5)(x^2 + 3x + 2)$
# Explanation:
## Step 1: Distribute $x$ and $5$ over the trinomial
$x(x^2 + 3x + 2) + 5(x^2 + 3x + 2)$

## Step 2: Multiply each term
$x \cdot x^2 + x \cdot 3x + x \cdot 2 + 5 \cdot x^2 + 5 \cdot 3x + 5 \cdot 2$
$= x^3 + 3x^2 + 2x + 5x^2 + 15x + 10$

## Step 3: Combine like terms
$x^3 + (3x^2 + 5x^2) + (2x + 15x) + 10$
$= x^3 + 8x^2 + 17x + 10$

# Answer:
$x^3 + 8x^2 + 17x + 10$

### Problem 2(b): $(2x - 3)(4x^2 + 5x - 7)$
# Explanation:
## Step 1: Distribute $2x$ and $-3$ over the trinomial
$2x(4x^2 + 5x - 7) - 3(4x^2 + 5x - 7)$

## Step 2: Multiply each term
$2x \cdot 4x^2 + 2x \cdot 5x + 2x \cdot (-7) - 3 \cdot 4x^2 - 3 \cdot 5x - 3 \cdot (-7)$
$= 8x^3 + 10x^2 - 14x - 12x^2 - 15x + 21$

## Step 3: Combine like terms
$8x^3 + (10x^2 - 12x^2) + (-14x - 15x) + 21$
$= 8x^3 - 2x^2 - 29x + 21$

# Answer:
$8x^3 - 2x^2 - 29x + 21$

### Problem 2(c): $(2x + 5)^3$
# Explanation:
## Step 1: Expand $(2x + 5)^3$ as $(2x + 5)(2x + 5)(2x + 5)$
First, multiply $(2x + 5)(2x + 5)$:
$(2x + 5)(2x + 5) = 4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25$

## Step 2: Multiply the result by $(2x + 5)$
$(4x^2 + 20x + 25)(2x + 5)$
Distribute each term:
$4x^2(2x) + 4x^2(5) + 20x(2x) + 20x(5) + 25(2x) + 25(5)$
$= 8x^3 + 20x^2 + 40x^2 + 100x + 50x + 125$

## Step 3: Combine like terms
$8x^3 + (20x^2 + 40x^2) + (100x + 50x) + 125$
$= 8x^3 + 60x^2 + 150x + 125$

# Answer:
$8x^3 + 60x^2 + 150x + 125$

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QUESTION IMAGE

Question

Explanation:

Problem 1(a): \((x + 5)(x + 8)\)

Step 1: Apply the distributive property (FOIL method)

Step 2: Combine like terms

Step 1: Apply the distributive property (FOIL method)

Step 2: Combine like terms

Step 1: Apply the distributive property (FOIL method)

Step 2: Combine like terms

Answer:

Problem 1(b): \((3x + 2)(2x - 7)\)