Factoring Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve and complete this worksheet step by step. This is a factoring using GCF (Greatest Common Factor) worksheet, so we'll fill in the blanks and work through each example.

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🔹 Part 1: Definitions

Factoring – finding ________ (expressions that ________ together to give the original expression)
✔ Answer:
> Factoring – finding factors (expressions that multiply together to give the original expression)

Example: $2x$ and $3x$ are factors of $6x^2$, because $2x \cdot 3x = 6x^2$

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Greatest Common Factor (GCF) – the _________ number that divides evenly into the original numbers
✔ Answer:
> GCF – the largest number that divides evenly into the original numbers

Example: Find the GCF of 12 and 16
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 16: 1, 2, 4, 8, 16
- Common factors: 1, 2, 4 → GCF = 4

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GCF of monomials – to find the GCF of monomials, find the GCF of the ___________ and any common variables with the _________ exponent
✔ Answer:
> GCF of monomials – to find the GCF of monomials, find the GCF of the coefficients and any common variables with the lowest exponent

Example: Find the GCF of $15x^3$ and $20x^5$
- Coefficients: GCF of 15 and 20 → 5
- Variables: both have $x$, lowest exponent is 3 → $x^3$
→ GCF = $5x^3$

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🔹 Factoring Out the GCF Steps: $6x^2 + 18x^5$

Let’s go through the steps:

#### ✔ STEP 1: Find the GCF of all terms
- Terms: $6x^2$ and $18x^5$
- Coefficients: GCF of 6 and 18 → 6
- Variables: both have $x$, lowest exponent is 2 → $x^2$
- So, GCF = $6x^2$

#### ✔ STEP 2: Divide all terms by the GCF
- $6x^2 ÷ 6x^2 = 1$
- $18x^5 ÷ 6x^2 = 3x^3$

#### ✔ STEP 3: Put the GCF outside parentheses and remainder inside
- $6x^2(1 + 3x^3)$

✔ Check by distributing:
- $6x^2 \cdot 1 = 6x^2$
- $6x^2 \cdot 3x^3 = 18x^5$
→ Correct!

So, final answer:
$$
\boxed{6x^2(1 + 3x^3)}
$$

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🔹 More Examples

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#### 1. $15y^3 + 25y^2$

- Coefficients: GCF of 15 and 25 → 5
- Variables: both have $y$, lowest exponent is 2 → $y^2$
- GCF = $5y^2$

Divide:
- $15y^3 ÷ 5y^2 = 3y$
- $25y^2 ÷ 5y^2 = 5$

Factor:
$$
5y^2(3y + 5)
$$

✔ Check: $5y^2(3y) = 15y^3$, $5y^2(5) = 25y^2$ → ✔️

Answer: $\boxed{5y^2(3y + 5)}$

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#### 2. $9x^3 - 3x^2 + 27x$

- Coefficients: 9, 3, 27 → GCF = 3
- Variables: all terms have $x$, lowest exponent is 1 → $x$
- GCF = $3x$

Divide:
- $9x^3 ÷ 3x = 3x^2$
- $-3x^2 ÷ 3x = -x$
- $27x ÷ 3x = 9$

Factor:
$$
3x(3x^2 - x + 9)
$$

✔ Check: $3x(3x^2) = 9x^3$, $3x(-x) = -3x^2$, $3x(9) = 27x$ → ✔️

Answer: $\boxed{3x(3x^2 - x + 9)}$

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#### 3. $7xy^2 + 49x^2y$

- Coefficients: 7 and 49 → GCF = 7
- Variables:
- Both have $x$: lowest exponent = 1 → $x$
- Both have $y$: lowest exponent = 1 → $y$
- GCF = $7xy$

Divide:
- $7xy^2 ÷ 7xy = y$
- $49x^2y ÷ 7xy = 7x$

Factor:
$$
7xy(y + 7x)
$$

✔ Check: $7xy(y) = 7xy^2$, $7xy(7x) = 49x^2y$ → ✔️

Answer: $\boxed{7xy(y + 7x)}$

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#### 4. $12x^5 + 36x^2y - 18y^3$

- Coefficients: 12, 36, 18 → GCF = 6
- Variables:
- $x^5$, $x^2$, and no $x$ in last term → no common $x$
- $y$ appears in two terms but not all → no common $y$
- So, only common factor is 6

GCF = 6

Divide:
- $12x^5 ÷ 6 = 2x^5$
- $36x^2y ÷ 6 = 6x^2y$
- $-18y^3 ÷ 6 = -3y^3$

Factor:
$$
6(2x^5 + 6x^2y - 3y^3)
$$

✔ Check: $6 \cdot 2x^5 = 12x^5$, $6 \cdot 6x^2y = 36x^2y$, $6 \cdot (-3y^3) = -18y^3$ → ✔️

Answer: $\boxed{6(2x^5 + 6x^2y - 3y^3)}$

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✔ Final Completed Worksheet Summary

#### Definitions:
- Factoring – finding factors (expressions that multiply together to give the original expression)
- GCF – the largest number that divides evenly into the original numbers
- GCF of monomials – find GCF of coefficients and common variables with the lowest exponent

#### Example: GCF of $15x^3$ and $20x^5$
- GCF = $5x^3$

#### Factoring $6x^2 + 18x^5$:
- Step 1: GCF = $6x^2$
- Step 2: $1$ and $3x^3$
- Step 3: $6x^2(1 + 3x^3)$

#### More Examples:
1. $15y^3 + 25y^2 = \boxed{5y^2(3y + 5)}$
2. $9x^3 - 3x^2 + 27x = \boxed{3x(3x^2 - x + 9)}$
3. $7xy^2 + 49x^2y = \boxed{7xy(y + 7x)}$
4. $12x^5 + 36x^2y - 18y^3 = \boxed{6(2x^5 + 6x^2y - 3y^3)}$

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