Solutions on how factor polynomials by common factor are presented.
Use common factors to factor completely the following polynomials.
a) \[-3x + 9\] b) \[28x + 2x^2\] c) \[11xy + 55x^2y\] d) \[20xy + 35x^2y - 15xy^2\] e) \[5y(x + 1) + 10y^2(x + 1) - 15xy(x + 1)\]
Find any common factors in the two terms of \( -3x + 9 \) by expressing both terms, \( -3x \) and \( 9 \), using prime factorization.
\[ -3x + 9 = -\color{red}{3} \cdot x + \color{red}{3} \cdot 3 \] The greatest common factor is \( \color{red}{3} \), which is factored out. Hence, \[ -3x + 9 = \color{red}{3}(-x + 3) = -3(x - 3) \] b)Write the prime factorization of each of the terms in the given polynomial \( 28x + 2x^2 \).
\[ 28x + 2x^2 = \color{red}{2} \cdot 2 \cdot 7 \cdot \color{red}{x} + \color{red}{2} \cdot \color{red}{x} \cdot x \] The greatest common factor is \( \color{red}{2x} \), and it is factored out. Hence, \[ 28x + 2x^2 = \color{red}{2x}(14 + x) \] c)Write the prime factorization of each of the terms in the given polynomial \( 11xy + 55x^2y \). \[ 11xy + 55x^2y = \color{red}{11} \cdot \color{red}{x} \cdot \color{red}{y} + 5 \cdot \color{red}{11} \cdot \color{red}{x} \cdot x \cdot \color{red}{y} \] The greatest common factor is \( \color{red}{11xy} \), which is factored out. Hence, \[ 11xy + 55x^2y = \color{red}{11xy}(1 + 5x) \] d)
Write the prime factorization of each of the terms in the given polynomial \( 20xy + 35x^2y - 15xy^2 \). \[ 20xy + 35x^2y - 15xy^2 = 2 \cdot 2 \cdot \color{red}{5} \cdot \color{red}{x} \cdot \color{red}{y} + \color{red}{5} \cdot 7 \cdot \color{red}{x} \cdot x \cdot \color{red}{y} - 3 \cdot \color{red}{5} \cdot \color{red}{x} \cdot \color{red}{y} \cdot y \] The greatest common factor is \( \color{red}{5xy} \), and it is factored out. Hence, \[ 20xy + 35x^2y - 15xy^2 = \color{red}{5xy}(4 + 7x - 3y) \] e)
We start by factoring out the common factor \( (x + 1) \) in the given polynomial.
\[ 5y(x + 1) + 10y^2(x + 1) - 15xy(x + 1) = (x + 1)(5y + 10y^2 - 15xy) \]
We now factor the polynomial \( 5y + 10y^2 - 15xy \) using the greatest common factor (GCF) of all three terms.
\[ 5y + 10y^2 - 15xy = \color{red}{5 \cdot y} + 2 \cdot \color{red}{5 \cdot y} \cdot y - 3 \cdot \color{red}{5 \cdot y} \cdot x = \color{red}{5 \cdot y}(1 + 2y - 3x) \]
The given polynomial may be factored as follows:
\[ 5y(x + 1) + 10y^2(x + 1) - 15xy(x + 1) = 5y(x + 1)(1 + 2y - 3x) \]