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1. Remove the greatest common factor and express as the product of a GCF and a polynomial.
a. [tex]7b^2 + 7b[/tex]
b. [tex]24ab - 18ab^2[/tex]
c. [tex]5a^2b^2 - 35ab[/tex]
d. [tex]8x^3 - 10x^2 - 14x[/tex]
e. [tex]36x^6 + 45x^4 - 18x^2[/tex]
f. [tex]40a^2 - 16ab - 24a[/tex]
g. [tex]8x^3y^2 + 32xy^2[/tex]
h. [tex]y(3x - 2) - (3x - 2)[/tex]
i. [tex]5x(x - 7) + 3(x - 7)[/tex]
j. [tex]4b(3ab - 2) + 2(3ab - 2) - 5a(3ab - 2)[/tex]

2. Factor by Grouping
a. [tex]xy - 2x + 7y - 14[/tex]
b. [tex]x^3 - 8x^2 + 3x - 24[/tex]
c. [tex]ad + 3a - d^2 - 3d[/tex]
d. [tex]xy + 5x - 5y - 25[/tex]
e. [tex]35 - 5a + 7a - a^2[/tex]

Sagot :

GinnyAnswer
Sure! Let's tackle each part step-by-step.

### 1. Remove the greatest common factor (GCF) and express each expression as the product of a GCF and a polynomial.

#### a. [tex]\(7 b^2 + 7 b\)[/tex]
1. Identify the GCF: The GCF of [tex]\(7 b^2\)[/tex] and [tex]\(7 b\)[/tex] is [tex]\(7 b\)[/tex].
2. Factor out the GCF: [tex]\(7 b(b + 1)\)[/tex]

So, the expression is [tex]\(7 b(b + 1)\)[/tex].

#### b. [tex]\(24 a b - 18 a b^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(24 a b\)[/tex] and [tex]\(18 a b^2\)[/tex] is [tex]\(6 a b\)[/tex].
2. Factor out the GCF: [tex]\(6 a b(4 - 3 b)\)[/tex]

So, the expression is [tex]\(6 a b(4 - 3 b)\)[/tex].

#### c. [tex]\(5 a^2 b^2 - 35 a b\)[/tex]
1. Identify the GCF: The GCF of [tex]\(5 a^2 b^2\)[/tex] and [tex]\(35 a b\)[/tex] is [tex]\(5 a b\)[/tex].
2. Factor out the GCF: [tex]\(5 a b(a b - 7)\)[/tex]

So, the expression is [tex]\(5 a b(a b - 7)\)[/tex].

#### d. [tex]\(8 x^3 - 10 x^2 - 14 x\)[/tex]
1. Identify the GCF: The GCF of [tex]\(8 x^3\)[/tex], [tex]\(10 x^2\)[/tex], and [tex]\(14 x\)[/tex] is [tex]\(2 x\)[/tex].
2. Factor out the GCF: [tex]\(2 x(4 x^2 - 5 x - 7)\)[/tex]

So, the expression is [tex]\(2 x(4 x^2 - 5 x - 7)\)[/tex].

#### e. [tex]\(36 x^6 + 45 x^4 - 18 x^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(36 x^6\)[/tex], [tex]\(45 x^4\)[/tex], and [tex]\(18 x^2\)[/tex] is [tex]\(9 x^2\)[/tex].
2. Factor out the GCF: [tex]\(9 x^2(4 x^4 + 5 x^2 - 2)\)[/tex]

So, the expression is [tex]\(9 x^2(4 x^4 + 5 x^2 - 2)\)[/tex].

#### f. [tex]\(40 a^2 - 16 a b - 24 a\)[/tex]
1. Identify the GCF: The GCF of [tex]\(40 a^2\)[/tex], [tex]\(16 a b\)[/tex], and [tex]\(24 a\)[/tex] is [tex]\(8 a\)[/tex].
2. Factor out the GCF: [tex]\(8 a(5 a - 2 b - 3)\)[/tex]

So, the expression is [tex]\(8 a(5 a - 2 b - 3)\)[/tex].

#### g. [tex]\(8 x^3 y^2 + 32 x y^2\)[/tex]
1. Identify the GCF: The GCF of [tex]\(8 x^3 y^2\)[/tex] and [tex]\(32 x y^2\)[/tex] is [tex]\(8 x y^2\)[/tex].
2. Factor out the GCF: [tex]\(8 x y^2(x^2 + 4)\)[/tex]

So, the expression is [tex]\(8 x y^2(x^2 + 4)\)[/tex].

#### h. [tex]\(y(3 x - 2) - (3 x - 2)\)[/tex]
1. Identify the GCF: The GCF of [tex]\(y(3 x - 2)\)[/tex] and [tex]\(-(3 x - 2)\)[/tex] is [tex]\((3 x - 2)\)[/tex].
2. Factor out the GCF: [tex]\((3 x - 2)(y - 1)\)[/tex]

So, the expression is [tex]\((3 x - 2)(y - 1)\)[/tex].

#### i. [tex]\(5 x(x - 7) + 3(x - 7)\)[/tex]
1. Identify the GCF: The GCF of [tex]\(5 x(x - 7)\)[/tex] and [tex]\(3(x - 7)\)[/tex] is [tex]\((x - 7)\)[/tex].
2. Factor out the GCF: [tex]\((x - 7)(5 x + 3)\)[/tex]

So, the expression is [tex]\((x - 7)(5 x + 3)\)[/tex].

#### j. [tex]\(4 b(3 a b - 2) + 2(3 a b - 2) - 5 a(3 a b - 2)\)[/tex]
1. Identify the GCF: The GCF of all three terms is [tex]\((3 a b - 2)\)[/tex].
2. Factor out the GCF: [tex]\((3 a b - 2)(4 b + 2 - 5 a)\)[/tex]

So, the expression is [tex]\((3 a b - 2)(4 b + 2 - 5 a)\)[/tex].

### 2. Factor by Grouping

#### a. [tex]\(x y - 2 x + 7 y - 14\)[/tex]
1. Group terms: [tex]\( (x y - 2 x) + (7 y - 14) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x(y - 2) + 7(y - 2) \)[/tex]
3. Factor out the common term [tex]\((y - 2)\)[/tex]: [tex]\( (x + 7)(y - 2) \)[/tex]

So, the expression is [tex]\((x + 7)(y - 2)\)[/tex].

#### b. [tex]\(x^3 - 8 x^2 + 3 x - 24\)[/tex]
1. Group terms: [tex]\( (x^3 - 8 x^2) + (3 x - 24) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x^2(x - 8) + 3(x - 8) \)[/tex]
3. Factor out the common term [tex]\((x - 8)\)[/tex]: [tex]\( (x^2 + 3)(x - 8) \)[/tex]

So, the expression is [tex]\((x^2 + 3)(x - 8)\)[/tex].

#### c. [tex]\(a d + 3 a - d^2 - 3 d\)[/tex]
1. Group terms: [tex]\( (a d + 3 a) - (d^2 + 3 d) \)[/tex]
2. Factor out the GCF from each group: [tex]\( a(d + 3) - d(d + 3) \)[/tex]
3. Factor out the common term [tex]\((d + 3)\)[/tex]: [tex]\( (a - d)(d + 3) \)[/tex]

So, the expression is [tex]\((a - d)(d + 3)\)[/tex].

#### d. [tex]\(x y + 5 x - 5 y - 25\)[/tex]
1. Group terms: [tex]\( (x y + 5 x) - (5 y + 25) \)[/tex]
2. Factor out the GCF from each group: [tex]\( x(y + 5) - 5(y + 5) \)[/tex]
3. Factor out the common term [tex]\((y + 5)\)[/tex]: [tex]\( (x - 5)(y + 5) \)[/tex]

So, the expression is [tex]\((x - 5)(y + 5)\)[/tex].

#### e. [tex]\(35 - 5 a + 7 a - a^2\)[/tex]
1. Rearrange and group terms: [tex]\( (-a^2 + 7 a) + (35 - 5 a) \rightarrow (7 a - a^2) + (35 - 5 a) \)[/tex]
2. Factor out the GCF from each group: [tex]\( a(7 - a) + 5(7 - a) \)[/tex]
3. Factor out the common term [tex]\((7 - a)\)[/tex]: [tex]\( (a + 5)(7 - a) \)[/tex]

So, the expression is [tex]\( (a + 5)(7 - a) \)[/tex].

That's all. These are the factored forms of each given expression.
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