Question

factor by grouping. (4 points each)

1. $x^3 + 8x^2 + 2x + 16$ \t\t\t\t\t2) $x^3 - 5x^2 - 3x + 15$

factor by the ac method. (6 points each)

1. $3x^2 - 2x - 5$ \t\t\t\t\t\t\t4) $2x^2 + 5x + 2$

1. $10x^2 - 3x - 1$ \t\t\t\t\t\t\t6) $5x^2 - 18x + 9$

Explanation:

Problem 1: Factor \( x^3 + 8x^2 + 2x + 16 \) by Grouping

Step 1: Group Terms

Group the first two terms and the last two terms:
\( (x^3 + 8x^2) + (2x + 16) \)

Step 2: Factor Out GCF from Each Group

Factor \( x^2 \) from the first group and \( 2 \) from the second group:
\( x^2(x + 8) + 2(x + 8) \)

Step 3: Factor Out Common Binomial

Factor out \( (x + 8) \) from both terms:
\( (x + 8)(x^2 + 2) \)

Step 1: Group Terms

Group the first two terms and the last two terms:
\( (x^3 - 5x^2) + (-3x + 15) \)

Step 2: Factor Out GCF from Each Group

Factor \( x^2 \) from the first group and \( -3 \) from the second group:
\( x^2(x - 5) - 3(x - 5) \)

Step 3: Factor Out Common Binomial

Factor out \( (x - 5) \) from both terms:
\( (x - 5)(x^2 - 3) \)

Step 1: Identify \( a, b, c \)

For \( ax^2 + bx + c \), here \( a = 3 \), \( b = -2 \), \( c = -5 \).

Step 2: Compute \( ac \)

\( ac = 3(-5) = -15 \).

Step 3: Find Two Numbers for \( ac \)

Find two numbers that multiply to \( -15 \) and add to \( b = -2 \).
Numbers: \( -5 \) and \( 3 \) (since \( -5 \times 3 = -15 \) and \( -5 + 3 = -2 \)).

Step 4: Split Middle Term

Rewrite \( -2x \) as \( -5x + 3x \):
\( 3x^2 - 5x + 3x - 5 \)

Step 5: Factor by Grouping

Group and factor:
\( (3x^2 - 5x) + (3x - 5) = x(3x - 5) + 1(3x - 5) = (3x - 5)(x + 1) \)

Answer:

\( \boldsymbol{(x + 8)(x^2 + 2)} \)

Problem 2: Factor \( x^3 - 5x^2 - 3x + 15 \) by Grouping