Factoring Simple Polynomials

Factoring is the reverse of multiplying. When you multiply \((x + 3)(x + 2)\), you get \(x^2 + 5x + 6\). When you factor \(x^2 + 5x + 6\), you're finding those original binomials: \((x + 3)(x + 2)\).

Factoring is useful for solving equations, simplifying expressions, and understanding the structure of polynomials. It's like reverse-engineering a recipe: instead of baking (multiplying), you're figuring out what ingredients went in (factoring).

There are several factoring techniques. This lesson covers the most fundamental two: factoring out the greatest common factor (GCF) and factoring by grouping.

Finding the Greatest Common Factor

The GCF is the largest expression that divides evenly into all terms of a polynomial. To factor out the GCF, you pull it out front and divide each term by it.

Example: Factor \(6x + 9\).

Both 6 and 9 are divisible by 3. So 3 is the GCF.

$$6x + 9 = 3(2x + 3)$$

You can verify this by distributing: \(3(2x + 3) = 6x + 9\) ✓

Example: Factor \(4x^2 + 8x\).

The GCF of 4 and 8 is 4. Both terms also have at least one \(x\), so \(x\) is part of the GCF too.

GCF: \(4x\)

$$4x^2 + 8x = 4x(x + 2)$$

Check: \(4x(x + 2) = 4x^2 + 8x\) ✓

Example: Factor \(15y^3 - 10y^2 + 5y\).

The coefficients 15, 10, and 5 all have GCF of 5. All three terms have at least one \(y\).

GCF: \(5y\)

$$15y^3 - 10y^2 + 5y = 5y(3y^2 - 2y + 1)$$

Check: \(5y(3y^2 - 2y + 1) = 15y^3 - 10y^2 + 5y\) ✓

Quick check: what is the GCF of \(12x^3 + 8x^2\)? Show answerThe GCF of 12 and 8 is 4. Both terms have at least \(x^2\). So the GCF is \(4x^2\), and the factored form is \(4x^2(3x + 2)\).

When There's a Negative GCF

Sometimes it's helpful to factor out a negative number if the leading coefficient is negative.

Example: Factor \(-6x - 12\).

You could factor out 6: \(6(-x - 2)\)

But it's often cleaner to factor out -6: \(-6(x + 2)\)

Both are correct, but the second form is more standard because the leading coefficient inside the parentheses is positive.

Example: Factor \(-3a^2 + 9a\).

Factor out \(-3a\):

$$-3a^2 + 9a = -3a(a - 3)$$

Factoring by Grouping

When a polynomial has four terms, you can sometimes factor it by grouping pairs of terms.

Example: Factor \(ax + ay + bx + by\).

Group the first two terms and the last two terms:

$$(ax + ay) + (bx + by)$$

Factor out the GCF from each group:

$$= a(x + y) + b(x + y)$$

Now notice that \((x + y)\) appears in both terms. Factor it out:

$$= (x + y)(a + b)$$

Example: Factor \(6x^2 + 9x + 4x + 6\).

Group: $$(6x^2 + 9x) + (4x + 6)$$

Factor GCF from each group: $$= 3x(2x + 3) + 2(2x + 3)$$

Factor out \((2x + 3)\): $$= (2x + 3)(3x + 2)$$

Check by multiplying back out. Using FOIL on \((2x + 3)(3x + 2)\):

  • First: \(6x^2\)
  • Outer: \(4x\)
  • Inner: \(9x\)
  • Last: \(6\)

Total: \(6x^2 + 4x + 9x + 6 = 6x^2 + 13x + 6\). That matches the original (after combining \(9x + 4x = 13x\)), so the factoring is correct.

Example: Factor \(2x^3 - 6x^2 + 5x - 15\).

Group: $$(2x^3 - 6x^2) + (5x - 15)$$

Factor each group: $$= 2x^2(x - 3) + 5(x - 3)$$

Factor out \((x - 3)\): $$= (x - 3)(2x^2 + 5)$$

When Grouping Doesn't Work Right Away

Sometimes you need to rearrange terms to make grouping work.

Example: Factor \(xy + 3y + 2x + 6\).

Grouping the first two and last two terms: $$(xy + 3y) + (2x + 6)$$ $$= y(x + 3) + 2(x + 3)$$ $$= (x + 3)(y + 2)$$

That worked. Rearranging the terms first also works: $$xy + 2x + 3y + 6$$ $$(xy + 2x) + (3y + 6)$$ $$= x(y + 2) + 3(y + 2)$$ $$= (y + 2)(x + 3)$$

Same result, just in a different order.

Combining GCF with Other Factoring

Always factor out the GCF first before trying other factoring methods.

Example: Factor \(3x^2 + 12x + 12\).

GCF is 3: $$= 3(x^2 + 4x + 4)$$

The trinomial \(x^2 + 4x + 4\) factors further, but trinomial factoring is an Algebra II topic. Stopping here gives the GCF factorization.

Checking Your Factoring

Always check by multiplying the factors back together. If you get the original polynomial, you factored correctly.

Example: Is \(5a(2a - 3)\) the correct factoring of \(10a^2 - 15a\)?

Check by distributing: \(5a(2a - 3) = 10a^2 - 15a\) ✓

Yes — the factoring is correct.

Give These a Try

Factor completely:

  1. \(12x + 18\) Show answerGCF is 6: \(6(2x + 3)\)

  2. \(5y^2 - 10y\) Show answerGCF is \(5y\): \(5y(y - 2)\)

  3. \(-4m^2 + 8m\) Show answerGCF is \(-4m\): \(-4m(m - 2)\). Factoring out a negative makes the leading term inside positive.

  4. \(ax + bx + ay + by\) Show answerGroup first two and last two terms: \(x(a + b) + y(a + b) = (a + b)(x + y)\)

  5. \(3x^2 + 6x + 2x + 4\) Show answerGroup: \((3x^2 + 6x) + (2x + 4) = 3x(x + 2) + 2(x + 2) = (x + 2)(3x + 2)\)

  6. \(8n^3 - 12n^2 + 10n - 15\) Show answerGroup: \((8n^3 - 12n^2) + (10n - 15) = 4n^2(2n - 3) + 5(2n - 3) = (2n - 3)(4n^2 + 5)\)

What's Next?

A few patterns worth keeping in mind. Always factor the GCF completely: \(8x^2 + 12x = 2x(4x + 6)\) isn't finished because the \(4x + 6\) inside still has a GCF of 2; the complete factoring is \(4x(2x + 3)\). The GCF includes the variable part too, so \(6x^3 + 9x^2\) has GCF \(3x^2\), not just 3. Watch the signs when factoring out a negative: \(-x + 5 = -1(x - 5)\), not \(-1(x + 5)\) — every term flips. Always verify by multiplying your factors back together and confirming you get the original. And not every polynomial factors nicely; if a GCF and grouping both fail, the polynomial might simply be prime, which is a valid answer.

The next step is full trinomial factoring (covered in Algebra II), which handles expressions like \(x^2 + 5x + 6\) directly. The GCF and grouping techniques here are usually the first move regardless of what comes next.

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