Question

A) (2a+3b)(4a²-6ab+9b²) b) (5-2y)(25x²+10xy+4y²) c) (m²-m+2)(3m²-2) d) (5-2a+a²)(4a²-3a-1)

Answer 1

a) (2a+3b)(4a²-6ab+9b²)=(2a)³+(3b)³=8a³+27b³

b) (5x-2y)(25x²+10xy+4y²)=(5x)³-(2y)³=125x³-8y³

c) (m²-m+2)(3m²-2)=3m⁴+4m³-3m³+2m-4

d) (5-2a+a²)(4a²-3a-1)=4a⁴-11a³+25a²-13a-5

Answer 2

$a)8 {a}^{3} - 12 {a}^{2} b + 18 {ab}^{2} + 12 {a}^{2} b \\ - 18 {ab}^{2} + 27 {b}^{3} = 8 {a}^{3}$+27b^3
$b)125 {x}^{2} + 50xy + 20 {y}^{2} - 50 {x}^{2} y \\ - 20 {xy}^{2} - 8 {y}^{3}$
$c)3 {m}^{4} - 2 {m}^{2} - {3m}^{3} + 2m \\ + {6m}^{2} - 4 = 3 {m}^{4} + {4m}^{2} - {3m}^{3} \\ + 2m$
$d)20 {a}^{2} - 15a - 5 - 8 {a}^{3} + 6 {a}^{2} \\ + 2a + 4 {a}^{4} - 3 {a}^{3} - {a}^{2} = \\ 25{a}^{2} - 13a - 5 - 11 {a}^{3} + 4 {a}^{4}$

Answer 3

To simplify each of the given expressions, you can apply the distributive property of multiplication over addition. This means that you will multiply each term in the first set of parentheses by each term in the second set of parentheses. Let's solve each expression step by step:

a) (2a + 3b)(4a² - 6ab + 9b²)

1. Multiply the first term in the first set of parentheses (2a) by each term in the second set of parentheses: 2a * 4a² = 8a³ 2a * (-6ab) = -12a²b 2a * 9b² = 18ab²

2. Multiply the second term in the first set of parentheses (3b) by each term in the second set of parentheses: 3b * 4a² = 12a²b 3b * (-6ab) = -18ab² 3b * 9b² = 27b³

Now, combine the like terms: Final expression: 8a³ - 12a²b + 12a²b - 18ab² + 18ab² + 27b³

Notice that the terms "-12a²b" and "+12a²b" cancel each other out, and "-18ab²" and "+18ab²" also cancel each other out. So the final simplified expression is:

Simplified expression: 8a³ + 27b³

b) (5 - 2y)(25x² + 10xy + 4y²)

1. Multiply the first term in the first set of parentheses (5) by each term in the second set of parentheses: 5 * 25x² = 125x² 5 * 10xy = 50xy 5 * 4y² = 20y²

2. Multiply the second term in the first set of parentheses (-2y) by each term in the second set of parentheses: -2y * 25x² = -50x²y -2y * 10xy = -20xy² -2y * 4y² = -8y³

Now, combine the like terms: Final expression: 125x² + 50xy - 50x²y + 20y² - 20xy² - 8y³

c) (m² - m + 2)(3m² - 2)

1. Multiply the first term in the first set of parentheses (m²) by each term in the second set of parentheses: m² * 3m² = 3m⁴ m² * (-2) = -2m²

2. Multiply the second term in the first set of parentheses (-m) by each term in the second set of parentheses: -m * 3m² = -3m³ -m * (-2) = 2m

3. Multiply the third term in the first set of parentheses (2) by each term in the second set of parentheses: 2 * 3m² = 6m² 2 * (-2) = -4

Now, combine the like terms: Final expression: 3m⁴ - 2m² - 3m³ + 2m + 6m² - 4

Rearrange the terms in descending order of powers of m: Final simplified expression: 3m⁴ - 3m³ + 4m² + 2m - 4

d) (5 - 2a + a²)(4a² - 3a - 1)

1. Multiply the first term in the first set of parentheses (5) by each term in the second set of parentheses: 5 * 4a² = 20a² 5 * (-3a) = -15a 5 * (-1) = -5

2. Multiply the second term in the first set of parentheses (-2a) by each term in the second set of parentheses: -2a * 4a² = -8a³ -2a * (-3a) = 6a² -2a * (-1) = 2a

3. Multiply the third term in the first set of parentheses (a²) by each term in the second set of parentheses: a² * 4a² = 4a⁴ a² * (-3a) = -3a³ a² * (-1) = -a²

Now, combine the like terms: Final expression: 4a⁴ - 8a³ + 20a² - 3a³ + 6a² - 15a - a² - 5

Rearrange the terms in descending order of powers of a: Final simplified expression: 4a⁴ - 8a³ - 3a³ + 20a² + 6a² - a² - 15a - 5

Combine like terms again: Final simplified expression: 4a⁴ - 11a³ + 25a² - 15a - 5