A tank is bought at the pet store and is known to have a volume of 12 cubic feet. The dimensions are shown in the diagram below. If your new pet requires the tank to be at least 3 feet high, did you buy a big enough tank?
Watch This
Khan Academy Factoring by Grouping
Note: The above video shows factoring by grouping of quadratic (trinomial) expressions. The same problem-solving concept will be developed in this lesson for cubic polynomials.
Guidance
Recall that to factor means to rewrite an expression as a product. In general, quadratic expressions are the easiest to factor and cubic expressions are much more difficult to factor.
One method that can be used to factor some cubics is the factoring by grouping method. To factor cubic polynomials by grouping there are four steps:
- Step 1: Separate the terms into two groups.
- Step 2: Factor out the common terms in each of the two groups.
- Step 3: Factor out the common binomial.
- Step 4: If possible, factor the remaining quadratic expression.
Take a look at the examples to see what factoring by grouping looks like.
Example A
Factor the following polynomial by grouping: \begin{align*}w^3-2w^2-9w+18\end{align*}
Solution: Step 1: Separate the terms into two groups. Notice the sign change on the second group because of the negative sign.
\begin{align*}w^3-2w^2-9w+18=(w^3-2w^2)-(9w-18)\end{align*}
Step 2: Factor out the common terms in each of the sets of parentheses.
\begin{align*}(w^3-2w^2)-(9w-18) = w^2(w-2)-9(w-2)\end{align*}
Step 3: Factor out the common binomial \begin{align*}(w - 2)\end{align*}
\begin{align*}w^2(w-2)-9(w-2) = (w-2)(w^2-9)\end{align*}
Step 4: Factor the remaining quadratic expression \begin{align*}(w^2-9)\end{align*}
\begin{align*}(w-2)(w^2-9) = (w-2)(w+3)(w-3)\end{align*}
Therefore, your answer is: \begin{align*}w^3-2w^2-9w+18 = (w-2)(w+3)(w-3)\end{align*}
Example B
Factor the following polynomial by grouping: \begin{align*}2s^3-8s^2+3s-12\end{align*}
Solution: Step 1: Separate the terms into two groups.
\begin{align*}2s^3-8s^2+3s-12 = (2s^3-8s^2)+(3s-12)\end{align*}
Step 2: Factor out the common terms in each of the sets of parentheses.
\begin{align*}(2s^3-8s^2) + (3s-12) = 2s^2(s-4)+3(s-4)\end{align*}
Step 3: Factor out the common binomial \begin{align*}(s-4)\end{align*}
\begin{align*}2s^2(s-4) + 3(s-4) = (s-4) (2s^2+3)\end{align*}
Step 4: Check to see if the remaining quadratic can be factored. In this case, the expression \begin{align*}(2s^3+3)\end{align*}
Therefore, your final answer is \begin{align*}2s^3-8s^2+3s-12 = (s-4)(2s^2+3)\end{align*}
Example C
Factor the following polynomial by grouping: \begin{align*}y^3+5y^2-4y-20\end{align*}
Solution: Step 1: Separate the terms into two groups. Notice the sign change on the second group because of the negative sign.
\begin{align*}y^3+5y^2-4y-20=(y^3+5y^2)-(4y+20)\end{align*}
Step 2: Factor out the common terms in each of the sets of parentheses.
\begin{align*}(y^3+5y^2)-(4y+20) = y^2(y+5)-4(y+5)\end{align*}
Step 3: Factor out the common binomial \begin{align*}(y + 5)\end{align*}.
\begin{align*}y^2(y+5)-4(y+5)=(y+5)(y^2-4)\end{align*}
Step 4: Factor the remaining quadratic expression \begin{align*}(y^2-4)\end{align*}.
\begin{align*}(y+5)(y^2-4)=(y+5)(y+2)(y-2)\end{align*}
Therefore, your answer is \begin{align*}y^3+5y^2-4y-20=(y+5)(y+2)(y-2)\end{align*}.
Concept Problem Revisited
A tank is bought at the pet store and is known to have a volume of 12 cubic feet. The dimensions are shown in the diagram below. If your new pet requires the tank to be at least 3 feet high, did you buy a big enough tank?
To solve this problem, you need to calculate the volume of the tank.
\begin{align*}V &= l\times w\times h \\ 12 &= (x+4) \times (x-1) \times (x) \\ 12 &= (x^2 +3x -4) \times (x) \\ 12 &= x^3 + 3x^2 -4x \\ 0 &= x^3 + 3x^2 -4x -12\end{align*}
Now you start to solve by factoring by grouping.
\begin{align*}0 = (x^3 + 3x^2) - (4x + 12)\end{align*}
Factor out the common terms in each of the sets of parentheses.
\begin{align*}0 = x^2 (x+3) - 4(x + 3)\end{align*}
Factor out the group of terms \begin{align*}(x + 3)\end{align*} from the expression.
\begin{align*}0 = (x+3)(x^2-4)\end{align*}
Completely factor the remaining quadratic expression.
\begin{align*}0 = (x+3)(x-2)(x+2)\end{align*}
Now solve for the variable \begin{align*}x\end{align*}.
\begin{align*}& \qquad 0 = (x+3) \ (x-2) \ (x+2) \\ & \qquad \qquad \swarrow \qquad \quad \ \downarrow \qquad \quad \ \searrow \\ & \quad \ \ x+3=0 \quad x-2=0 \quad x+2=0 \\ & \qquad \quad \ x=-3 \qquad x=2 \qquad \ \ x=-2 \end{align*}
Since you are looking for a length, only \begin{align*}x = 2\end{align*} is a good solution (you can't have a negative length!). But since you need a tank 3 feet high and this one is only 2 feet high, you need to go back to the pet shop and buy a bigger one.
Vocabulary
- Cubic Polynomial
- A cubic polynomial is a polynomial of degree equal to 3. For example \begin{align*}8x^3+2x^2-5x-7\end{align*} is a cubic polynomial.
- Distributive Property
- The distributive property states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. For example, the distributive property says that \begin{align*}2(x+5)=2x+10\end{align*}.
- Factor
- To factor means to rewrite an expression as a product of other expressions. These resulting expressions are called the factors of the original expression.
- Factor Completely
- To factor completely means to factor an expression until none of its factors can be factored any further.
- Greatest Common Factor
- The Greatest Common Factor (or GCF) is the largest monomial that is a factor of (or divides into evenly) each of the terms of the polynomial.
Guided Practice
Factor each of the following polynomials by grouping.
1. Factor the following polynomial by grouping: \begin{align*}y^3-4y^2-4y+16\end{align*}.
2. Factor the following polynomial by grouping: \begin{align*}3x^3-4x^2-3x+4\end{align*}.
3. Factor the following polynomial by grouping: \begin{align*}e^3+3e^2-4e-12\end{align*}.
Answers
1. Here are the steps:
\begin{align*}y^3-4y^2-4y+16&=y^2(y-4)-4(y-4)\\&=(y^2-4)(y-4)\\&=(y-2)(y+2)(y-4)\end{align*}
2. Here are the steps:
\begin{align*}3x^3-4x^2-3x+4&=x^2(3x-4)-1(3x-4)\\&=(x^2-1)(3x-4)\\&=(x-1)(x+1)(3x-4)\end{align*}
3. Here are the steps:
\begin{align*}e^3+3e^2-4e-12&=e^2(e+3)-4(e+3)\\&=(e^2-4)(e+3)\\&=(e+2)(e-2)(e+3)\end{align*}
Practice
Factor the following cubic polynomials by grouping.
- \begin{align*}x^3-3x^2-36x+108\end{align*}
- \begin{align*}e^3-3e^2-81e+243\end{align*}
- \begin{align*}x^3-10x^2-49x+490\end{align*}
- \begin{align*}y^3-7y^2-5y+35\end{align*}
- \begin{align*}x^3+9x^2+3x+27\end{align*}
- \begin{align*}3x^3+x^2-3x-1\end{align*}
- \begin{align*}5s^3-6s^2-45s+54\end{align*}
- \begin{align*}4a^3-7a^2+4a-7\end{align*}
- \begin{align*}5y^3+15y^2-45y-135\end{align*}
- \begin{align*}3x^3+15x^2-12x-60\end{align*}
- \begin{align*}2e^3+14e^2+7e+49\end{align*}
- \begin{align*}2k^3+16k^2+38k+24\end{align*}
- \begin{align*}-6x^3+3x^2+54x-27\end{align*}
- \begin{align*}-5m^3-6m^2+20m+24\end{align*}
- \begin{align*}-2x^3-8x^2+14x+56\end{align*}