Section 7.2 Graphing Polynomial Functions
Subsection 1. Factor polynomials
Factoring can help us analyze a polynomial. In particular, for polynomials in one variable, factoring may help us find the \(x\)-intercepts of the graph.
Subsubsection Examples
Example 7.13.
Factor completely \(~4x^3y-12x^2y-40xy\)
First, factor out the common factor, \(~4xy\text{.}\)
Now factor the quadratic trinomial. We need two numbers \(p\) and \(q\) that satisfy
By trial and error we find \(~p=-5~\) and \(~q=2~\text{,}\) so
and thus
Example 7.14.
Factor the quadratic trinomial \(~8x^2-21x-9\)
We want to find two factors so that \(~8x^2-21x-9 = (ax+b)(cx+d)\text{.}\) Now
so \(~(ad)(bc)=8(-9)=-72~\) and \(~ad+bc=-21.~\)
To simplify the calculations, we let \(~p=ad~\) and \(~q=bc~\text{.}\) We need to find the two numbers \(p\) and \(q\) that satisfy
By trying different factors of \(-72\text{,}\) we find that \(~p=-24~\) and \(~q=3~\text{.}\)
Finally, we write the trinomial as \(~8x^2-24x+3x-9~\text{,}\) and factor by grouping:
Thus, \(~8x^2-21x-9 = (x-3)(8x+3).\)
Subsubsection Exercises
Checkpoint 7.15.
Factor completely \(~18a^2b-9ab-27b\)
Checkpoint 7.16.
Factor completely \(~4x^3+12x^2y+8xy^2\)
Checkpoint 7.17.
Factor completely \(~9x^3y+9x^2y^2-18xy^3\)
Checkpoint 7.18.
Factor completely \(~12b^3y^2+15b^2y+3b\)
Checkpoint 7.19.
Factor completely \(~5x^2-14x-24\)
Checkpoint 7.20.
Factor completely \(~12t^2-10t-50\)
Subsection 2. Factor special products
Here is more practice using formulas to factor the special quadratic and cubic polynomials.
Quadratic Polynomials.
Cubic Polynomials.
Subsubsection Examples
Example 7.21.
Factor \(~x^4-24x^2+144\)
From the square terms we see that \(a=x^2\) and \(b=12\text{.}\) We check that the middle term is \(-2ab\text{.}\)
The polynomial fits the pattern for \((a-b)^2\text{,}\) so
Example 7.22.
Factor \(~27t^3+8v^2\)
The polynomial is the sum of two cubes, with \(x=3t\) and \(y=2v\text{.}\) We substitute these values into the formula.
Thus,
Subsubsection Exercises
Checkpoint 7.23.
Factor \(~a^6-4a^3b+4b^2\)
Checkpoint 7.24.
Factor \(~m^2+30m+225\)
Checkpoint 7.25.
Factor \(~125a^{12}+1\)
Checkpoint 7.26.
Factor \(~64p^3-q^6\)
Subsection 3. Divide polynomials
If a polynomial cannot be factored, we can use a process like long division to write it as a quotient plus a remainder.
Subsubsection Example
Example 7.27.
Divide \(~\dfrac{2x^2+x-7}{x+3}\)
First write
and divide \(2x^2\) (the first term of the numerator) by \(x\) (the first term of the denominator) to obtain \(x\text{.}\) (It may be helpful to write down the division: \(\dfrac{2x^2}{x}=2x\text{.}\)) Write \(\alert{2x}\) above the quotient bar as the first term of the quotient, as shown below.
Next, multiply \(x+3\) by \(2x\) to obtain \(2x^2+6x\text{,}\) and subtract this product from \(2x^2+x-7\text{.}\)
Repeating the process, divide \(-5x\) by \(x\) to obtain \(-5\text{.}\) Write \(-5\) as the second term of the quotient. Then multiply \(x+3\) by \(-5\) to obtain \(-5x-15\text{,}\) and subtract:
The remainder is 8. Because the degree of 8 is less than the degree of \(x+3\text{,}\) the division is finished. The quotient is \(2x-5\text{,}\) with a remainder of 8. We write the remainder as a fraction to obtain
Subsubsection Exercises
Checkpoint 7.28.
Divide \(~\dfrac{4x^2+12x+7}{2x+1}\)
Checkpoint 7.29.
Divide \(~\dfrac{8z^4+4z^2+5z+3}{2z+1}\)