Section 4.6 Factoring Binomials
¶Difference of Squares.
A binomial (two term polynomial) of form \(a^2-b^2\) always factors into the product \((a+b)(a-b)\text{.}\) We can confirm this by applying FOIL to the expression \((a+b)(a-b)\text{.}\)
A few simple examples follow. As always, we can check our result by expanding the factored expression.
Now let's consider a few expressions that don't immediately fit the pattern. Consider \(x^{10}-16\text{.}\) Hopefully we are quick to see that \(16\) is the square of \(4\text{.}\) To use our factor pattern successfully, we need to also recognize that \(x^{10}\) is a perfect square, as is any even power of \(x\text{.}\) The power-to-a-power rule of exponents relates that \((x^m)^n=x^{mn}\text{.}\) So the power of \(x\) we square that results in \(x^{10}\) must be half of \(10\text{,}\) i.e. \(5\text{.}\) Putting it all together we have:
Similar examples follow.
Exploration 4.6.1.
Exploration 4.6.2.
Sum and Difference of Cubes.
A binomial (two term polynomial) of form \(a^3-b^3\) always factors into the product \((a-b)(a^2+ab+b^2)\text{.}\) We can confirm this by expanding the expression \((a-b)(a^2+ab+b^2)\text{.}\)
Similarly, a binomial of form \(a^3+b^3\) always factors into the product \((a+b)(a^2-ab+b^2)\text{.}\) We can confirm this by expanding the expression \((a+b)(a^2-ab+b^2)\text{.}\)
Example 4.6.3.
Factor \(8x^3+27\) and \(8x^3-27\text{.}\)
For both binomials, \(8x^3\) corresponds to what is identified in the patterns as \(a^3\) and \(27\) corresponds to what is identified in the pattern as \(b\text{.}\) The resultant expressions for \(a\text{,}\) \(b\text{,}\) \(a^2\text{,}\) and \(b^2\) and shown in Figure 4.6.4 and the factorizations are shown to the left of the table.
| \(a^3=8x^3\) | \(b^3=27\) |
| \(a=2x\) | \(b=3\) |
| \(a^2=4x^2\) | \(b^2=9\) |
| \(ab=6x\) |
Example 4.6.5.
Factor the binomials \(1+64x^{15}\) and \(1-64x^{15}\text{.}\)
Note that the power-to-a-power rule of exponents gives us \((x^5)^3=x^{(5\times 3)}\text{.}\)
| \(a^3=1\) | \(b^3=64x^{15}\) |
| \(a=1\) | \(b=4x^5\) |
| \(a^2=1\) | \(b^2=16x^{10}\) |
| \(ab=4x^5\) |
Exploration 4.6.3.
Exploration 4.6.4.
Sum of Squares.
Unless the expression also happens to be a sum of cubes, sums of squares do not factor - that is, they are prime.
\(x^2+4\) is prime.
\(y^4+25\) is prime.
\(w^6+4x^2\) is prime.
Many folks would like \(x^2+4\) to factor, so much so that they will write \(x^2+4=(x+2)^2\text{.}\) Would that it were so. But alas:
In summary, \(x^2+4\neq (x+2)^2\text{,}\) \(x^2+4x+4=(x+2)^2\text{.}\)
Exercises Exercises
Factor each binomial after first completing the indicated table.
1.
Use the factor pattern \(a^2-b^2=(a+b)(a-b)\) to factor \(x^{10}-25y^4\) after first completing the entries in Figure 4.6.9
| \(a^2=\) | \(b^2=\) |
| \(a=\) | \(b=\) |
\(x^{10}-25y^4=(x^5+5y^2)(x^5-5y^2)\)
| \(a^2=x^{10}\) | \(b^2=25y^4\) |
| \(a=x^5\) | \(b=5y^2\) |
| \(ab=5x^5y^2\) |
2.
Use the factor pattern \(a^3-b^3=(a-b)(a^2+ab+b^2)\) to factor \(8x^3-y^6\) after first completing the entries in Figure 4.6.11
| \(a^3=\) | \(b^3=\) |
| \(a=\) | \(b=\) |
| \(a^2=\) | \(b^2=\) |
| \(ab=\) |
\(8x^3-y^6=(2x-y^2)(4x^2+2xy^2+y^4)\)
| \(a^3=8x^3\) | \(b^3=y^6\) |
| \(a=2x\) | \(b=y^2\) |
| \(a^2=4x^2\) | \(b^2=y^4\) |
| \(ab=2xy^2\) |
3.
Use the factor pattern \(a^3+b^3=(a+b)(a^2-ab+b^2)\) to factor \(125t^{12}+27x^9\) after first completing the entries in Figure 4.6.13
| \(a^3=\) | \(b^3=\) |
| \(a=\) | \(b=\) |
| \(a^2=\) | \(b^2=\) |
| \(ab=\) |
| \(a^3=125t^{12}\) | \(b^3=27x^9\) |
| \(a=5t^4\) | \(b=3x^3\) |
| \(a^2=25t^8\) | \(b^2=9x^6\) |
| \(ab=15t^4x^3\) |
\(125t^{12}+27x^9=(5t^4+3x^3)(25t^8-15t^4x^3+9x^6)\)
Factor each binomial. Check your result by expanding the factored expression. If the binomial does not factor, state that it is prime.