Solve Simple Polynomial Equations
Learning Outcomes
- Greatest Common Factor
- Review the concept of greatest common factor
- Factor a Polynomial
- Simple Polynomial Equations
- Factor the greatest common monomial out of a polynomial
- Solve a polynomial in factored form by setting it equal to zero
Factor Tree for 54[latex]t\left(5-t\right)=0[/latex]
But because the equation was written as the product of two terms, we could use the zero product principle. What if we are given a polynomial equation that is not written as a product of two terms, such as this one [latex]2y^2+4y=0[/latex]? We can use a technique called factoring, where we try to find factors that can be divided into each term of the polynomial so it can be rewritten as a product.
In this section we will explore how to find common factors from the terms of a polynomial, and rewrite it as a product. This technique will help us solve polynomial equations in the next section.
Greatest Common Factor
When we studied fractions, we learned that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 4 is the GCF of 16 and 20 because it is the largest number that divides evenly into both 16 and 20.The GCF of polynomials works the same way: 4x is the GCF of 16x and [latex]20x^2[/latex] because it is the largest polynomial that divides evenly into both 16x and [latex]20x^2[/latex].A General Note: Greatest Common Factor
The greatest common factor (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.
Prime Numbers[latex]\begin{array}{l}50=10\cdot5\\30=10\cdot3\end{array}[/latex]
Their greatest common factor is 10, since 10 is the greatest factor that both numbers have in common. To find the GCF of greater numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together.Example
Find the greatest common factor of 210 and 168.Answer:
[latex]\begin{array}{l}\,\,\,\,210=2\cdot3\cdot5\cdot7\\\,\,\,\,168=2\cdot2\cdot2\cdot3\cdot7\\\text{GCF}=2\cdot3\cdot7\end{array}[/latex]
Answer
[latex-display]\text{GCF}=42[/latex-display]Example
Find the greatest common factor of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex].Answer:
[latex]\begin{array}{l}\,\,25b^{3}=5\cdot5\cdot{b}\cdot{b}\cdot{b}\\\,\,10b^{2}=5\cdot2\cdot{b}\cdot{b}\\\text{GCF}=5\cdot{b}\cdot{b}\end{array}[/latex]
Answer
[latex-display]\text{GCF}=5b^{2}[/latex-display]Example
Find the greatest common factor of [latex]81c^{3}d[/latex] and [latex]45c^{2}d^{2}[/latex].Answer:
[latex]\begin{array}{l}\,\,\,81c^{3}d=3\cdot3\cdot3\cdot3\cdot{c}\cdot{c}\cdot{c}\cdot{d}\\45c^{2}d^{2}=3\cdot3\cdot5\cdot{c}\cdot{c}\cdot{d}\cdot{d}\\\,\,\,\,\text{GCF}=3\cdot3\cdot{c}\cdot{c}\cdot{d}\end{array}[/latex]
Answer
[latex-display]\text{GCF}=9c^{2}d[/latex-display]Factor a Polynomial
One of these things is not like the others.| Polynomial | Terms | Common Factors |
|---|---|---|
| [latex]6x+9[/latex] | 6x and 9 | 3 is a factor of 6x and 9 |
| [latex]a^{2}–2a[/latex] | [latex]a^{2}[/latex] and [latex]−2a[/latex] | a is a factor of [latex]a^{2}[/latex] and [latex]−2a[/latex] |
| [latex]4c^{3}+4c[/latex] | [latex]4c^{3}[/latex] and [latex]4c[/latex] | 4 and c are factors of [latex]4c^{3}[/latex] and 4c |
Distributive Property Forward and Backward
Forward: Product of a number and a sum: [latex]a\left(b+c\right)=a\cdot{b}+a\cdot{c}[/latex]. You can say that “[latex]a[/latex] is being distributed over [latex]b+c[/latex].” Backward: Sum of the products: [latex]a\cdot{b}+a\cdot{c}=a\left(b+c\right)[/latex]. Here you can say that “a is being factored out.” We first learned that we could distribute a factor over a sum or difference, now we are learning that we can "undo" the distributive property with factoring.Example
Factor [latex]25b^{3}+10b^{2}[/latex].Answer: Find the GCF. From a previous example, you found the GCF of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex] to be [latex]5b^{2}[/latex].
[latex]\begin{array}{l}\,\,25b^{3}=5\cdot5\cdot{b}\cdot{b}\cdot{b}\\\,\,10b^{2}=5\cdot2\cdot{b}\cdot{b}\\\text{GCF}=5\cdot{b}\cdot{b}=5b^{2}\end{array}[/latex]
Rewrite each term with the GCF as one factor.[latex]\begin{array}{l}25b^{3} = 5b^{2}\cdot5b\\10b^{2}=5b^{2}\cdot2\end{array}[/latex]
Rewrite the polynomial using the factored terms in place of the original terms.[latex]5b^{2}\left(5b\right)+5b^{2}\left(2\right)[/latex]
Factor out the [latex]5b^{2}[/latex].[latex]5b^{2}\left(5b+2\right)[/latex]
Answer
[latex-display]5b^{2}\left(5b+2\right)[/latex-display][latex]\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5.\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}.\end{array}[/latex]
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.Example
Factor [latex]81c^{3}d+45c^{2}d^{2}[/latex].Answer: Factor [latex]81c^{3}d[/latex].
[latex]3\cdot3\cdot9\cdot{c}\cdot{c}\cdot{c}\cdot{d}[/latex]
Factor [latex]45c^{2}d^{2}[/latex].[latex]3\cdot3\cdot5\cdot{c}\cdot{c}\cdot{d}\cdot{d}[/latex]
Find the GCF.[latex]3\cdot3\cdot{c}\cdot{c}\cdot{d}=9c^{2}d[/latex]
Rewrite each term as the product of the GCF and the remaining terms.[latex]\begin{array}{l}\,\,81c^{3}d=9c^{2}d\left(9c\right)\\45c^{2}d^{2}=9c^{2}d\left(5d\right)\end{array}[/latex]
Rewrite the polynomial expression using the factored terms in place of the original terms.[latex]9c^{2}d\left(9c\right)+9c^{2}d\left(5d\right)[/latex]
Factor out [latex]9c^{2}d[/latex].[latex]9c^{2}d\left(9c+5d\right)[/latex]
Answer
[latex-display]9c^{2}d\left(9c+5d\right)[/latex-display]Simple Polynomial Equations
In this section we will apply factoring a monomial from a polynomial to solving polynomial equations. Recall that not all of the techniques we use for solving linear equations will apply to solving polynomial equations, so we will be using the zero product principle to solve for a variable. We will begin with an example where the polynomial is already equal to zero.Example
Solve: [latex-display]-t^2+t=0[/latex-display]Answer: Each term has a common factor of t, so we can factor and use the zero product principle. Rewrite each term as the product of the GCF and the remaining terms. [latex-display]\begin{array}{c}-t^2=t\left(-t\right)\\t=t\left(1\right)\end{array}[/latex-display] Rewrite the polynomial equation using the factored terms in place of the original terms.
[latex]\begin{array}{c}-t^2+t=0\\t\left(-t\right)+t\left(1\right)\\t\left(-t+1\right)=0\end{array}[/latex]
Now we have a product on one side and zero on the other, so we can set each factor equal to zero using the zero product principle.
[latex]\begin{array}{c}t=0\,\,\,\,\,\,\,\,\text{ OR }\,\,\,\,\,\,\,\,\,\,\,-t+1=0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-1}\,\,\,\underline{-1}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-t=-1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{-t}{-1}=\frac{-1}{-1}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,t=1\end{array}[/latex]
Answer
[latex-display]t=0\text{ OR }t=1[/latex-display]Example
Solve: [latex]6t=3t^2-12t[/latex]Answer: First, move all the terms to one side. The goal is to try and see if we can use the zero product principle, since that is the only tool we know for solving polynomial equations.
[latex]\begin{array}{c}\,\,\,\,\,\,\,6t=3t^2-12t\\\underline{-6t}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-6t}\\\,\,\,\,\,\,\,0=3t^2-18t\\\end{array}[/latex]
We now have all the terms on the right side, and zero on the other side. Each term has a common factor of 3t, so we can factor and use the zero product principle. If you are uncertain how we found the common factor 3t, review the section on greatest common factor.
Rewrite each term as the product of the GCF and the remaining terms. Note how we leave the negative sign on the 6 when we factor 3t out of -18t.
[latex]\begin{array}{c}3t^2=3t\left(t\right)\\-18t=3t\left(-6\right)\end{array}[/latex]
Rewrite the polynomial equation using the factored terms in place of the original terms.
[latex]\begin{array}{c}0=3t^2-18t\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0=3t\left(t\right)+3t\left(-6\right)\\0=3t\left(t-6\right)\end{array}[/latex]
Solve the two equations.
[latex]\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,0=t-6\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ OR }\,\,\,\,\,\,\,\,\,0=3t\\\,\,\,\,\,\,\,\,\,\underline{+6}\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{+6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{0}{3}=\frac{3t}{3}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,6=t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ OR }\,\,\,\,\,\,\,\,\,0=t\end{array}[/latex]
Answer
[latex-display]t=6\text{ OR }t=0[/latex-display]Example
Solve [latex]\frac{1}{2}y=-4y-\frac{1}{2}y^2[/latex]Answer: We can solve this in one of two ways. One way is to eliminate the fractions like you may have done when solving linear equations, and the second is to find a common denominator and factor fractions. We will work through the second way. First, we will find a common denominator, then factor fractions.
[latex]\begin{array}{l}\frac{1}{2}y=-4y-\frac{1}{2}y^2\\0=-\frac{1}{2}y-4y-\frac{1}{2}y^2\end{array}[/latex]
2 is a common denominator for [latex]-\frac{1}{2}y\text{ and }-4y[/latex][latex]\frac{2}{2}\cdot{-4y}=-\frac{8y}{2}[/latex]
Rewrite the equation with the common denominator, then combine like terms.[latex]\begin{array}{l}0=-\frac{1}{2}y-4y-\frac{1}{2}y^2\\\text{ }\\0=-\frac{1}{2}y-\frac{8y}{2}-\frac{1}{2}y^2\\\text{}\\0=-\frac{9}{2}y-\frac{1}{2}y^2\end{array}[/latex]
Find the greatest common factor of the terms of the polynomial:
Factors of [latex]-\frac{9}{2}y[/latex]
[latex]-\frac{1}{2}\cdot{3}\cdot{3}\cdot{y}[/latex]
Factors of [latex]-\frac{1}{2}y^2[/latex]
[latex]-\frac{1}{2}\cdot{y}\cdot{y}[/latex]
Both terms have [latex]-\frac{1}{2}\text{ and }y[/latex] in common.
Rewrite each term as the product of the GCF and the remaining terms.
[latex]-\frac{9}{2}y=-\frac{1}{2}y\left(3\cdot{3}\right)=-\frac{1}{2}y\left(9\right)[/latex]
[latex]-\frac{1}{2}y^2=-\frac{1}{2}y\left(y\right)[/latex]
Rewrite the polynomial equation using the factored terms in place of the original terms. Pay attention to signs when we factor. Notice that we end up with a sum as a factor because the common factor is a negative number. [latex]\left(9+y\right)[/latex]
[latex]\begin{array}{l}0=-\frac{9}{2}y-\frac{1}{2}y^2\\\text{}\\-\frac{1}{2}y\left(9\right)-\frac{1}{2}y\left(y\right)\\\text{}\\-\frac{1}{2}y\left(9+y\right)\end{array}[/latex]
Solve the two equations.
[latex]\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,0=9+y\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ OR }\,\,\,\,\,\,\,\,\,0=-\frac{1}{2}y\\\,\,\,\,\,\,\,\,\,\underline{-9}\,\,\,\,\,\,\underline{-9}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\,\,\,\,\,\,\,\,\,-9=y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ OR }\,\,\,\,\,\,\,\,\,0=y\end{array}[/latex]
Answer
[latex]y=-9\text{ OR }y=0[/latex]
- We needed a common denominator to combine the like terms [latex]-4y\text{ and }-\frac{1}{2}y[/latex], after we moved all the terms to one side of the equation
- We found the GCF of the terms [latex]-\frac{9}{2}y\text{ and }-\frac{1}{2}y^2[/latex]
- We used the GCF to factor the polynomial [latex]-\frac{9}{2}y-\frac{1}{2}y^2[/latex]
- We used the zero product principle to solve the polynomial equation [latex]0=-\frac{1}{2}y\left(9+y\right)[/latex]
If the GCF of a polynomial is negative, pay attention to the signs that are left when you factor it from the terms of a polynomial.Summary
A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\cdot{b}+a\cdot{c}[/latex] form has been rewritten as [latex]a\left(b+c\right)[/latex], where a is the GCF, the polynomial is in factored form. In this section we practiced using the zero product principle as a method for solving polynomial equations. We found that the techniques we used to solve linear equations did not give us the correct answer when used to solve a polynomial equation. We also found that a polynomial can be rewritten as a product by factoring out the greatest common factor. We used both factoring and the zero product principle to solve second degree polynomials.Licenses & Attributions
CC licensed content, Original
- Screenshot One of These Things. Provided by: Lumen Learning License: CC BY: Attribution.
- Screenshot Factor Tree. Provided by: Lumen Learning License: CC BY: Attribution.
- Screenshot Prime Numbers. Provided by: Lumen Learning License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Factor and Solve a Quadratic Equation - GCF Only. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Factor and Solve a Quadratic Equation with Fractions - GCF Only . Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Example: Determining the Greatest Common Factor. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Ex: Determine the GCF of Two Monomials (One Variables). Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Ex: Determine the GCF of Two Monomials (Two Variables). Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Ex 1: Identify GCF and Factor a Binomial. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Ex 1: Factor and Solve a Quadratic Equation - GCF . Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
