Question

polynomials and factoring - review assignment
(d) $(2c + 6)(3c^3 - 6c + 8)$

1. find the area of the shaded region.

image of a square with side $3x + 2$ and a smaller square inside with top side $x - 2$ and right side $x + 1$

1. factor (divide out) the greatest common factor (gcf) from each polynomial.

(a) $2x^2y^3 - 14xy^3$
(b) $35m^2np - 15m^3n^2 + 30m^2n^2p$

Explanation:

Problem 4: Find the area of the shaded region

Step 1: Area of outer square

The outer figure is a square with side length \( 3x + 2 \). The area of a square is \( \text{side}^2 \), so the area of the outer square is \( (3x + 2)^2 \).
Expanding \( (3x + 2)^2 \) using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we get \( (3x)^2 + 2(3x)(2) + 2^2 = 9x^2 + 12x + 4 \).

Step 2: Area of inner rectangle

The inner figure is a rectangle with length \( x - 2 \) and width \( x + 1 \). The area of a rectangle is \( \text{length} \times \text{width} \), so the area of the inner rectangle is \( (x - 2)(x + 1) \).
Expanding \( (x - 2)(x + 1) \) using the distributive property (FOIL method), we get \( x^2 + x - 2x - 2 = x^2 - x - 2 \).

Step 3: Area of shaded region

The area of the shaded region is the area of the outer square minus the area of the inner rectangle. So we subtract the two areas:
\( (9x^2 + 12x + 4) - (x^2 - x - 2) \)
Simplify by combining like terms:
\( 9x^2 + 12x + 4 - x^2 + x + 2 = 8x^2 + 13x + 6 \)

Step 1: Find GCF of coefficients and variables

• For the coefficients (2 and 14), the GCF is 2.
• For the variable \( x \): the lowest power of \( x \) in the terms is \( x^1 \) (from \( 14xy^3 \)).
• For the variable \( y \): the lowest power of \( y \) in the terms is \( y^3 \) (both terms have \( y^3 \)).

So the GCF is \( 2xy^3 \).

Step 2: Divide each term by GCF

Divide \( 2x^2y^3 \) by \( 2xy^3 \): \( \frac{2x^2y^3}{2xy^3} = x \)
Divide \( -14xy^3 \) by \( 2xy^3 \): \( \frac{-14xy^3}{2xy^3} = -7 \)

Step 3: Write factored form

Factor out the GCF from the polynomial: \( 2xy^3(x - 7) \)

Step 1: Find GCF of coefficients and variables

• For the coefficients (35, 15, 30), the GCF is 5.
• For the variable \( m \): the lowest power of \( m \) in the terms is \( m^2 \) (from \( 35m^2np \) and \( 30m^2n^2p \)).
• For the variable \( n \): the lowest power of \( n \) in the terms is \( n^1 \) (from \( 35m^2np \)).
• For the variable \( p \): The term \( -15m^3n^2 \) has no \( p \), so the lowest power of \( p \) is \( p^0 = 1 \) (we don't include \( p \) in the GCF).

So the GCF is \( 5m^2n \).

Step 2: Divide each term by GCF

Divide \( 35m^2np \) by \( 5m^2n \): \( \frac{35m^2np}{5m^2n} = 7p \)
Divide \( -15m^3n^2 \) by \( 5m^2n \): \( \frac{-15m^3n^2}{5m^2n} = -3mn \)
Divide \( 30m^2n^2p \) by \( 5m^2n \): \( \frac{30m^2n^2p}{5m^2n} = 6np \)

Step 3: Write factored form

Factor out the GCF from the polynomial: \( 5m^2n(7p - 3mn + 6np) \)

Answer:

The area of the shaded region is \( 8x^2 + 13x + 6 \)

Problem 5(a): Factor out the GCF from \( 2x^2y^3 - 14xy^3 \)