Comprehensive Tutorial with Step-by-Step Solutions
Factoring a polynomial means writing it as a product of simpler polynomials. This is a fundamental skill in algebra used for solving equations and simplifying complex expressions.
Factoring by a common factor is based on using the Distributive Law in reverse:
Standard Distribution:
\[ a(b + c) = a \cdot b + a \cdot c \]Reverse (Factoring):
\[ a \cdot b + a \cdot c = a(b + c) \]In this method, we identify a factor $a$ that is common to all terms and move it outside the parentheses.
Note: To verify your factorization, simply expand the result to see if you return to the original polynomial.
Identify the Greatest Common Factor (GCF) and factor the following polynomials completely.
Example A: \( 9x - 6 \)
Solution: Use prime factorization to find common factors: \( \color{red}{3} \cdot 3 \cdot x - 2 \cdot \color{red}{3} \). The GCF is 3. Result: \( 3(3x - 2) \).
Example B: \( 16x^3 + 8x^2y + 4xy^2 \)
Solution: The GCF of the coefficients (16, 8, 4) is 4. The shared variable is $x$. The GCF is $4x$. Result: \( 4x(4x^2 + 2xy + y^2) \).
Example C: \( 2x^4(x + 5) + x^2(x + 5) \)
Solution: Notice $(x + 5)$ is a common binomial factor. Factor it out first: \( (x + 5)(2x^4 + x^2) \). Then factor $x^2$ from the remaining part. Result: \( x^2(x + 5)(2x^2 + 1) \).
Use common factors to factor completely the following polynomials. Each question includes a detailed solution.
Question 1: Factor \[ -3x + 9 \]
Express both terms using prime factorization:
\[ -3x + 9 = -\color{red}{3} \cdot x + \color{red}{3} \cdot 3 \]The greatest common factor is \( \color{red}{3} \). When factored out, we get:
\[ = \color{red}{3}(-x + 3) \]To have a positive leading term inside the parentheses, we can factor out $-3$:
\[ = -3(x - 3) \]Question 2: Factor \[ 28x + 2x^2 \]
Identify common factors through prime factorization:
\[ 28x + 2x^2 = \color{red}{2} \cdot 2 \cdot 7 \cdot \color{red}{x} + \color{red}{2} \cdot \color{red}{x} \cdot x \]The greatest common factor is \( \color{red}{2x} \):
\[ = \color{red}{2x}(14 + x) \text{ or } 2x(x + 14) \]Question 3: Factor \[ 11xy + 55x^2y \]
Write the prime factorization of each term:
\[ 11xy + 55x^2y = \color{red}{11} \cdot \color{red}{x} \cdot \color{red}{y} + 5 \cdot \color{red}{11} \cdot \color{red}{x} \cdot x \cdot \color{red}{y} \]The greatest common factor is \( \color{red}{11xy} \):
\[ = \color{red}{11xy}(1 + 5x) \]Question 4: Factor \[ 20xy + 35x^2y - 15xy^2 \]
Determine the GCF by analyzing all three terms:
\[ 20xy = 2^2 \cdot \color{red}{5} \cdot \color{red}{x} \cdot \color{red}{y} \] \[ 35x^2y = 7 \cdot \color{red}{5} \cdot \color{red}{x} \cdot x \cdot \color{red}{y} \] \[ -15xy^2 = -3 \cdot \color{red}{5} \cdot \color{red}{x} \cdot \color{red}{y} \cdot y \]The greatest common factor is \( \color{red}{5xy} \):
\[ = \color{red}{5xy}(4 + 7x - 3y) \]Question 5: Factor \[ 5y(x + 1) + 10y^2(x + 1) - 15xy(x + 1) \]
1. First, factor out the common binomial factor \( (x + 1) \):
\[ (x + 1)(5y + 10y^2 - 15xy) \]2. Now, identify the GCF for the remaining terms within the second parentheses ($5y, 10y^2, -15xy$). The GCF is \( \color{red}{5y} \):
\[ 5y + 10y^2 - 15xy = \color{red}{5y}(1 + 2y - 3x) \]3. Combine both steps for the final factored form:
\[ = 5y(x + 1)(1 + 2y - 3x) \]