Factorization of Polynomials worksheet for math students.

To solve the problem of factoring each polynomial, we will use the method of factoring quadratic expressions. The general form of a quadratic expression is \( ax^2 + bx + c \), and we aim to express it as \( (px + q)(rx + s) \).

Step-by-Step Solution:

#### 1. \( 6x^2 + 7x + 2 \)

We need to find two numbers that multiply to \( 6 \times 2 = 12 \) and add up to \( 7 \). These numbers are \( 3 \) and \( 4 \).

Rewrite the middle term using these numbers:
\[ 6x^2 + 7x + 2 = 6x^2 + 3x + 4x + 2 \]

Factor by grouping:
\[ = 3x(2x + 1) + 2(2x + 1) \]
\[ = (3x + 2)(2x + 1) \]

So, the correct answer is:
\[ \boxed{B} \]

#### 2. \( 2x^2 - 5x + 2 \)

We need to find two numbers that multiply to \( 2 \times 2 = 4 \) and add up to \( -5 \). These numbers are \( -4 \) and \( -1 \).

Rewrite the middle term using these numbers:
\[ 2x^2 - 5x + 2 = 2x^2 - 4x - x + 2 \]

Factor by grouping:
\[ = 2x(x - 2) - 1(x - 2) \]
\[ = (2x - 1)(x - 2) \]

So, the correct answer is:
\[ \boxed{D} \]

#### 3. \( 8x^2 + 18x - 5 \)

We need to find two numbers that multiply to \( 8 \times (-5) = -40 \) and add up to \( 18 \). These numbers are \( 20 \) and \( -2 \).

Rewrite the middle term using these numbers:
\[ 8x^2 + 18x - 5 = 8x^2 + 20x - 2x - 5 \]

Factor by grouping:
\[ = 4x(2x + 5) - 1(2x + 5) \]
\[ = (4x - 1)(2x + 5) \]

So, the correct answer is:
\[ \boxed{B} \]

#### 4. \( 9x^2 - 21x + 10 \)

We need to find two numbers that multiply to \( 9 \times 10 = 90 \) and add up to \( -21 \). These numbers are \( -15 \) and \( -6 \).

Rewrite the middle term using these numbers:
\[ 9x^2 - 21x + 10 = 9x^2 - 15x - 6x + 10 \]

Factor by grouping:
\[ = 3x(3x - 5) - 2(3x - 5) \]
\[ = (3x - 2)(3x - 5) \]

So, the correct answer is:
\[ \boxed{D} \]

#### 5. \( 6x^2 + 5x - 6 \)

We need to find two numbers that multiply to \( 6 \times (-6) = -36 \) and add up to \( 5 \). These numbers are \( 9 \) and \( -4 \).

Rewrite the middle term using these numbers:
\[ 6x^2 + 5x - 6 = 6x^2 + 9x - 4x - 6 \]

Factor by grouping:
\[ = 3x(2x + 3) - 2(2x + 3) \]
\[ = (3x - 2)(2x + 3) \]

So, the correct answer is:
\[ \boxed{A} \]

#### 6. \( 8x^2 - 2x - 15 \)

We need to find two numbers that multiply to \( 8 \times (-15) = -120 \) and add up to \( -2 \). These numbers are \( -12 \) and \( 10 \).

Rewrite the middle term using these numbers:
\[ 8x^2 - 2x - 15 = 8x^2 - 12x + 10x - 15 \]

Factor by grouping:
\[ = 4x(2x - 3) + 5(2x - 3) \]
\[ = (4x + 5)(2x - 3) \]

So, the correct answer is:
\[ \boxed{D} \]

#### 7. \( 4x^2 - 11x + 6 \)

We need to find two numbers that multiply to \( 4 \times 6 = 24 \) and add up to \( -11 \). These numbers are \( -8 \) and \( -3 \).

Rewrite the middle term using these numbers:
\[ 4x^2 - 11x + 6 = 4x^2 - 8x - 3x + 6 \]

Factor by grouping:
\[ = 4x(x - 2) - 3(x - 2) \]
\[ = (4x - 3)(x - 2) \]

So, the correct answer is:
\[ \boxed{C} \]

Final Answers:

1. \( \boxed{B} \)
2. \( \boxed{D} \)
3. \( \boxed{B} \)
4. \( \boxed{D} \)
5. \( \boxed{A} \)
6. \( \boxed{D} \)
7. \( \boxed{C} \)