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Section 4.6 Factoring Binomials

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Difference of Squares.

A binomial (two term polynomial) of form a2b2 always factors into the product (a+b)(ab). We can confirm this by applying FOIL to the expression (a+b)(ab).

(a+b)(ab)=a2ab+abb2=a2b2

A few simple examples follow. As always, we can check our result by expanding the factored expression.

x24=(x+2)(x2)
y225=(y+5)(y5)
36x2=(6+x)(6x)

Now let's consider a few expressions that don't immediately fit the pattern. Consider x1016. Hopefully we are quick to see that 16 is the square of 4. To use our factor pattern successfully, we need to also recognize that x10 is a perfect square, as is any even power of x. The power-to-a-power rule of exponents relates that (xm)n=xmn. So the power of x we square that results in x10 must be half of 10, i.e. 5. Putting it all together we have:

x1016=(x5+4)(x54)

Similar examples follow.

y89=(y4+3)(y43)
x461=(x23+1)(x231)
100w12=(10+w6)(10w6)
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Figure 4.6.1. Practice Factoring Difference of Squares with One Variable.
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Figure 4.6.2. Practice Factoring Difference of Squares with Two Variables.
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Sum and Difference of Cubes.

A binomial (two term polynomial) of form a3b3 always factors into the product (ab)(a2+ab+b2). We can confirm this by expanding the expression (ab)(a2+ab+b2).

(ab)(a2+ab+b2)=a3+a2b+ab2a2bab2b3=a3b3

Similarly, a binomial of form a3+b3 always factors into the product (a+b)(a2ab+b2). We can confirm this by expanding the expression (a+b)(a2ab+b2).

(a+b)(a2ab+b2)=a3a2b+ab2+a2bab2+b3=a3+b3
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Example 4.6.3.

Factor 8x3+27 and 8x327.

Solution

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.

\begin{gather*} 8x^3+27=(2x+3)(4x^2-6x+9)\\ \\ 8x^3-27=(2x-3)(4x^2+6x+9) \end{gather*}
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\(a^3=8x^3\) \(b^3=27\)
\(a=2x\) \(b=3\)
\(a^2=4x^2\) \(b^2=9\)
\(ab=6x\)
Figure 4.6.4.
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Example 4.6.5.

Factor the binomials 1+64x15 and 164x15.

Solution

Note that the power-to-a-power rule of exponents gives us \((x^5)^3=x^{(5\times 3)}\text{.}\)

\begin{gather*} 1+64x^{15}=(1+4x^5)(1-4x^5+16x^{10})\\ \\ 1-64x^{15}=(1-4x^5)(1+4x^5+16x^{10}) \end{gather*}
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\(a^3=1\) \(b^3=64x^{15}\)
\(a=1\) \(b=4x^5\)
\(a^2=1\) \(b^2=16x^{10}\)
\(ab=4x^5\)
Figure 4.6.6.
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Figure 4.6.7. Practice Factoring Sum or Difference of Cubes with one variable.
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Figure 4.6.8. Practice Factoring Sum or Difference of Cubes with two variables.
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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.

x2+4 is prime.

y4+25 is prime.

w6+4x2 is prime.

Many folks would like x2+4 to factor, so much so that they will write x2+4=(x+2)2. Would that it were so. But alas:

(x+2)2=(x+2)(x+2)=x2+2x+2x+4=x2+4x+4

In summary, x2+4(x+2)2, x2+4x+4=(x+2)2.

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Exercises Exercises

Factor each binomial after first completing the indicated table.

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1.

Use the factor pattern a2b2=(a+b)(ab) to factor x1025y4 after first completing the entries in Figure 4.6.9

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a2= b2=
a= b=
Figure 4.6.9.
Solution

\(x^{10}-25y^4=(x^5+5y^2)(x^5-5y^2)\)

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\(a^2=x^{10}\) \(b^2=25y^4\)
\(a=x^5\) \(b=5y^2\)
\(ab=5x^5y^2\)
Figure 4.6.10.
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2.

Use the factor pattern a3b3=(ab)(a2+ab+b2) to factor 8x3y6 after first completing the entries in Figure 4.6.11

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a3= b3=
a= b=
a2= b2=
ab=
Figure 4.6.11.
Solution

\(8x^3-y^6=(2x-y^2)(4x^2+2xy^2+y^4)\)

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\(a^3=8x^3\) \(b^3=y^6\)
\(a=2x\) \(b=y^2\)
\(a^2=4x^2\) \(b^2=y^4\)
\(ab=2xy^2\)
Figure 4.6.12.
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3.

Use the factor pattern a3+b3=(a+b)(a2ab+b2) to factor 125t12+27x9 after first completing the entries in Figure 4.6.13

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a3= b3=
a= b=
a2= b2=
ab=
Figure 4.6.13.
Solution
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\(a^3=125t^{12}\) \(b^3=27x^9\)
\(a=5t^4\) \(b=3x^3\)
\(a^2=25t^8\) \(b^2=9x^6\)
\(ab=15t^4x^3\)
Figure 4.6.14.

\(125t^{12}+27x^9=(5t^4+3x^3)(25t^8-15t^4x^3+9x^6)\)

Factor each binomial. Check your result by expanding the factored expression. If the binomial does not factor, state that it is prime.