Example

Factor [latex]6z^{2}+11z+4[/latex].

Answer: Rewrite the middle term, [latex]11z[/latex], as [latex]3z + 8z[/latex] (from the chart above.)

[latex]6z^{2}+3z+8z+4[/latex]

Group pairs. Use grouping to consider the terms in pairs.

[latex]\left(6z^{2}+3z\right)+\left(8z+4\right)[/latex]

Factor [latex]3[/latex]z out of the first group and [latex]4[/latex] out of the second group.

[latex]3z\left(2z+1\right)+4\left(2z+1\right)[/latex]

Factor out [latex]\left(2z+1\right)[/latex].

[latex]\left(2z+1\right)\left(3z+4\right)[/latex]

Answer

[latex-display]\left(2z+1\right)\left(3z+4\right)[/latex-display]

Example

Factor [latex]7x^{2}-16x–5[/latex].

Answer: Find [latex]p, q[/latex] such that [latex]p\cdot{q}=-35\text{ and }p+q=-16[/latex]

Factors of [latex]7\cdot{-5}=-35[/latex]	Sum of Factors
[latex]-1, 35[/latex]	[latex]34[/latex]
[latex]1, -35[/latex]	[latex]-34[/latex]
[latex]-5, 7[/latex]	[latex]2[/latex]
[latex]-7,5[/latex]	[latex]-2[/latex]

The trinomial cannot be factored. None of the factors add up to [latex]-16[/latex]

Example

Factor [latex]3x^{3}–3x^{2}–90x[/latex].

Answer: Since 3 is a common factor for the three terms, factor out the 3.

[latex]3\left(x^{3}–x^{2}–30x\right)[/latex]

x is also a common factor, so factor out x.

[latex]3x\left(x^{2}–x–30\right)[/latex]

Now you can factor the trinomial [latex]x^{2}–x–30[/latex]. To find r and s, identify two numbers whose product is [latex]−30[/latex] and whose sum is [latex]−1[/latex]. The pair of factors is [latex]−6[/latex] and [latex]5[/latex]. So replace [latex]–x[/latex] with [latex]−6x+5x[/latex].

[latex]3x\left(x^{2}–6x+5x–30\right)[/latex]

Use grouping to consider the terms in pairs.

[latex]3x\left[\left(x^{2}–6x\right)+\left(5x–30\right)\right][/latex]

Factor x out of the first group and factor [latex]5[/latex] out of the second group.

[latex]3x\left[\left(x\left(x–6\right)\right)+5\left(x–6\right)\right][/latex]

Then factor out [latex]x–6[/latex].

[latex]3x\left(x–6\right)\left(x+5\right)[/latex]

Answer

[latex-display]3x\left(x–6\right)\left(x+5\right)[/latex-display]

Example

Factor [latex]−4h^{2}+11h+3[/latex]

Answer: Factor [latex]−1[/latex] out of the trinomial. Notice that the signs of all three terms have changed.

[latex]−1\left(4h^{2}–11h–3\right)[/latex]

To factor the trinomial, you need to figure out how to rewrite [latex]−11h[/latex]. The product of [latex]rs=4\cdot−3=−12[/latex], and the sum of [latex]rs=−11[/latex].

[latex]r\cdot{s}=−12[/latex]	[latex]r+s=−11[/latex]
[latex]−12\cdot1=−12[/latex]	[latex]−12+1=−11[/latex]
[latex]−6\cdot2=−12[/latex]	[latex]−6+2=−4[/latex]
[latex]−4\cdot3=−12[/latex]	[latex]−4+3=−1[/latex]

Rewrite the middle term [latex]−11h[/latex] as [latex]−12h+1h[/latex].

[latex]−1\left(4h^{2}–12h+1h–3\right)[/latex]

Group terms.

[latex]−1\left[\left(4h^{2}–12h\right)+\left(1h–3\right)\right][/latex]

Factor out [latex]4[/latex]h from the first pair. The second group cannot be factored further, but you can write it as [latex]+1\left(h–3\right)[/latex] since [latex]+1\left(h–3\right)=\left(h–3\right)[/latex]. This helps with factoring in the next step.

[latex]−1\left[4h\left(h–3\right)+1\left(h–3\right)\right][/latex]

Factor out a common factor of [latex]\left(h–3\right)[/latex]. Notice you are left with [latex]\left(h–3\right)\left(4h+1\right)[/latex]; the [latex]+1[/latex] comes from the term [latex]+1\left(h–3\right)[/latex] in the previous step.

[latex]−1\left[\left(h–3\right)\left(4h+1\right)\right][/latex]

Answer

[latex-display]−1\left(h–3\right)\left(4h+1\right)[/latex-display]