1.7 Expanding and factoring
The form of an expression affects which features are easy to see and which operations are easy to perform.
Factor. An expression multiplied by another expression to produce a given product.
Factoring reverses expansion. Begin with the greatest factor \(A^2 - B^2\), then look for patterns such as
\[A^2-B^2=(A+B)(A-B).\]
Worked Example: Using a difference of squares
\[25x^2-9=(5x)^2-3^2=(5x-3)(5x+3).\]
Check Your Work
\((4y-7)(4y+7)\).Always check a factorization by multiplying the factors. A complete factorization removes the greatest common factor first and continues until none of the remaining factors can be factored over the number system in use. The perfect-square patterns are
\[A^2+2AB+B^2=(A+B)^2,\]
\[A^2-2AB+B^2=(A-B)^2.\]
The sum and difference of cubes are
\[A^3+B^3=(A+B)(A^2-AB+B^2),\]
\[A^3-B^3=(A-B)(A^2+AB+B^2).\]
Expressions containing negative or fractional powers can also be factored by removing the lowest power common to every term. This technique is useful later when simplifying derivatives and rate equations, even though those expressions are not polynomials.
1.7.1 A factoring decision process
First remove the greatest common factor. Next count the terms. Two terms may form a difference of squares or a sum or difference of cubes. Three terms may form a quadratic trinomial or perfect-square trinomial. Four terms may factor by grouping. After each step, inspect every resulting factor again.
For \(-7x^4y^2+14xy^3+21xy^4\), the greatest common factor is \(-7xy^2\). Factoring it out gives
\[-7xy^2(x^3-2y-3y^2).\]
Choosing a negative greatest common factor makes the leading term inside the parentheses positive. Expanding the result verifies the signs.
Worked Example: Removing a common flow factor
Factor \(12q^3-18q^2\). The greatest common factor is \(6q^2\):
\[12q^3-18q^2=6q^2(2q-3).\]
Check Your Work
\(5r^2(3r+2)\).For a trinomial with leading coefficient 1, such as \(x^2+9x-22\), find two numbers with product \(-22\) and sum 9. The values 11 and \(-2\) give \((x+11)(x-2)\). When the leading coefficient is not 1, the split-middle-term method avoids unsupported guessing. In \(4x^2-8x-21\), the required product is \(-84\) and the required sum is \(-8\). The pair 6 and \(-14\) leads to
\[ 4x^2+6x-14x-21=2x(2x+3)-7(2x+3)=(2x-7)(2x+3). \]
The factorization is confirmed by expansion. This last step is short and detects most sign or coefficient errors.
To factor \(ax^2+bx+c\), find two numbers whose product is \(ac\) and whose sum is \(b\), split the middle term, and group. For example,
\[6x^2+7x+2=(3x+2)(2x+1).\]
Worked Example: Factoring by grouping
For \(6x^2+7x+2\), the product \(ac\) is 12. The values 3 and 4 multiply to 12 and add to 7:
\[6x^2+3x+4x+2=3x(2x+1)+2(2x+1)=(3x+2)(2x+1).\]
Check Your Work
\((x+4)(x+5)\).Worked Example: Factoring a rectangular-area model
An expanded basin-area model is \(A=w^2+9w+20\). Factoring gives
\[A=(w+4)(w+5).\]
The factors recover the two dimensions represented by the model.
Check Your Work
\((d+3)(d+8)\).Applied Problem: Recovering basin dimensions. A rectangular basin has width \(x+2\) m and length \(x+7\) m. Its expanded area model is \(x^2+9x+14\). Factor the polynomial to verify the dimensions, then find the area when \(x=5\).
Check Your Work
\(x^2+9x+14=(x+2)(x+7)\). At \(x=5\), the dimensions are 7 m and 12 m, so the area is 84 m².1.7.2 Practice Problems
- Factor \(3x^2-6x\).
Check Your Work
\(3x(x-2)\). - Factor \(x^2-11x+24\).
Check Your Work
\((x-3)(x-8)\). - Factor \(6x^2+5x-4\).
Check Your Work
\((3x+4)(2x-1)\). - Factor \(9a^2-25\).
Check Your Work
\((3a-5)(3a+5)\). - Factor \(8x^3-1\).
Check Your Work
\((2x-1)(4x^2+2x+1)\). - Factor the pump-expression \(18q^2-30q\).
Check Your Work
\(6q(3q-5)\). - Factor the basin-area model \(w^2+13w+40\).
Check Your Work
\((w+5)(w+8)\). - Factor the difference between two squared pipe diameters, \(D^2-0.36\).
Check Your Work
\((D-0.6)(D+0.6)\).