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Factoring Completely

Sum or difference with higher powers, factoring algorithm, and grouping.

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Factoring Completely

Suppose that the area of a rectangular poster hanging on your wall is 1000 square inches and that the height is 15 more inches than the width. How would you go about finding the height and width of the poster? What equation would you set up? How could you use factoring to solve the equation?

Factoring Completely

We say that a polynomial is factored completely when we factor as much as we can and we are unable to factor any more. Here are some suggestions that you should follow to make sure that you factor completely.

\begin{align*}\checkmark\end{align*} Factor all common monomials first.

\begin{align*}\checkmark\end{align*} Identify special products such as the difference of squares or the square of a binomial. Factor according to their formulas.

\begin{align*}\checkmark\end{align*} If there are no special products, factor using the methods we learned in previous Concepts.

\begin{align*}\checkmark\end{align*} Look at each factor and see if any of these can be factored further.

Let's factor the following polynomials completely:

  1. \begin{align*}2x^2-8\end{align*}

Look for the common monomial factor: \begin{align*}2x^2-8=2(x^2-4)\end{align*}. Recognize \begin{align*}x^2-4\end{align*} as a difference of squares. We factor as follows: \begin{align*}2(x^2-4)=2(x+2)(x-2)\end{align*}. If we look at each factor we see that we can't factor anything else. The answer is \begin{align*}2(x+2)(x-2)\end{align*}.

  1. \begin{align*}x^3+6x^2+9x\end{align*}

Recognize this as a perfect square and factor as \begin{align*}x(x+3)^2\end{align*}. If we look at each factor we see that we can't factor anything else. The answer is \begin{align*}x(x+3)^2\end{align*}.

  1. \begin{align*}3x(x-1)+4(x-1)\end{align*}

\begin{align*}3x(x-1)+4(x-1)\end{align*} has a common binomial of \begin{align*}(x-1)\end{align*}.

When we factor the common binomial, we get \begin{align*}(x-1)(3x+4)\end{align*}.

Solving Real-World Problems Using Polynomial Equations

Now that we know most of the factoring strategies for quadratic polynomials, we can see how these methods apply to solving real-world problems.

Let's solve the following problem:

The product of two positive numbers is 60. Find the two numbers if one of the numbers is 4 more than the other.

\begin{align*}x=\end{align*} one of the numbers and \begin{align*}x+4\end{align*} equals the other number. The product of these two numbers equals 60. We can write the equation as follows:

\begin{align*}x(x+4)=60\end{align*}

Write the polynomial in standard form.

\begin{align*}x^2+4x&=60\\ x^2+4x-60&=0\end{align*}

Factor:

\begin{align*}-60=6 \times(-10)\end{align*} and \begin{align*}6+(-10)=-4\end{align*}

\begin{align*}-60=-6 \times 10\end{align*} and \begin{align*}-6+10=4\end{align*}

-6 and 10 are the correct factors. 

The expression factors as \begin{align*}(x+10)(x-6)=0\end{align*}.

Solve:

\begin{align*}x+10=0 && x-6& =0\\ \text{or} \\ x=-10 && x& =6\end{align*}

Since we are looking for positive numbers, the answer must be positive.

\begin{align*}x=6\end{align*} for one number, and \begin{align*}x+4=10\end{align*} for the other number.

Check: \begin{align*}6 \cdot 10=60\end{align*} so the answer checks.

   

 

 

 

Examples

Example 1

Earlier, you were told that the area of a rectangular poster hanging on you wall is 1000 square inches and that the height is 15 inches longer than the width. What is the height and width of the poster? What equation would you set up? 

The formula for finding the area of the poster is \begin{align*}length \times width = Area\end{align*}.

Let \begin{align*}w\end{align*} represent the width of the poster. The height is 15 inches longer than the width so the height is represented by the expression \begin{align*}w+15\end{align*}.  

Plugging this information into the formula from above:

\begin{align*}(w+15)(w) = 1000\end{align*} 

Note that we cannot set each of the two factors on the left equal to 1000 and solve. We need the right side of the equation to be equal to 0 in order to use the Zero Product Property.

Since we can't solve the equation in it's current form, we need to change it to standard form by distributing the left side and subtracting 1000 from both sides.

\begin{align*}(w+15)(w)&=1000\\ w^2 + 15w &=1000\\ w^2 + 15w -1000 &= 0\end{align*}

Now, we need to factor the left side of the equation.  

Note that this is not a special product. 1000 has many different factors but we know that the factors must add up to 15 so they must be close to each other. We know that one of the factors must be negative and one must be positive since \begin{align*}c\end{align*} is negative. Let's try out some of the pairs of factors:

\begin{align*}-1000 &= -200 \times 5 &&\text { and }&& -200 + 5 = -195 && \text {The numbers are too far apart}\\ -1000 &= -100 \times 10 &&\text { and }&& -100 + 10 = -90 && \text {The numbers are too far apart}\\ -1000 &= -50\times 20 &&\text { and }&& -50 + 20 = -30 && \text {The sum is getting closer to 15}\\ -1000 &= -40\times 25 &&\text { and }&& -40 + 25 = -15 \\ -1000 &= 40\times (-25) &&\text { and }&& 40 +(- 25) = 15 \end{align*}

The factors that multiply to -1000 and sum to 15 are 40 and -25.

Factoring, we get:

\begin{align*}w^2 + 15w -1000 &= 0\\ (w-25)(w+40) &=0\end{align*}

This equation is factored completely so we can set each factor equal to 0 and solve:

\begin{align*}w -25 = 0 && \text { and } && w+40 &= 0\\ w =25&& \text { and } && w &= -40\\ \end{align*}

Since we are talking about width, we cannot have a negative value. Therefore, the width of the poster is 25 inches. Since the height is 15 inches longer than the width, the height is 40 inches. 

Example 2

Factor completely: \begin{align*}24x^3-28x^2+8x\end{align*}.

First, notice that each term has \begin{align*}4x\end{align*} as a factor. Start by factoring out \begin{align*}4x\end{align*}:

\begin{align*}24x^3-28x^2+8x=4x(6x^2-7x+2)\end{align*}

Next, factor the trinomial in the parenthesis. Since \begin{align*}a\neq 1\end{align*} find \begin{align*}a\cdot c\end{align*}: \begin{align*} 6\cdot 2=12\end{align*}. Find the factors of 12 that add up to -7. Since 12 is positive and -7 is negative, the two factors should be negative:

\begin{align*} 12&=-1\cdot -12 && and && -1+-12=-13\\ 12&=-2\cdot -6 && and && -2+-6=-8\\ 12&=-3\cdot -4 && and && -3+-4=-7\end{align*}

Rewrite the trinomial using \begin{align*}-7x=-3x-4x\end{align*}, and then factor by grouping:

\begin{align*}6x^2-7x+2=6x^2-3x-4x+2=3x(2x-1)-2(2x-1)=(3x-2)(2x-1)\end{align*}

The final factored answer is:

\begin{align*}4x(3x-2)(2x-1)\end{align*}

Review

Factor completely.

  1. \begin{align*}2x^2+16x+30\end{align*}
  2. \begin{align*}12c^2-75\end{align*}
  3. \begin{align*}-x^3+17x^2-70x\end{align*}
  4. \begin{align*}6x^2-600\end{align*}
  5. \begin{align*}-5t^2-20t-20\end{align*}
  6. \begin{align*}6x^2+18x-24\end{align*}
  7. \begin{align*}-n^2+10n-21\end{align*}
  8. \begin{align*}2a^2-14a-16\end{align*}
  9. \begin{align*}2x^2-512\end{align*}
  10. \begin{align*}12x^3+12x^2+3x\end{align*}

Solve the following application problems.

  1. One leg of a right triangle is seven feet longer than the other leg. The hypotenuse is 13 feet. Find the dimensions of the right triangle.
  2. A rectangle has sides of \begin{align*}x+2\end{align*} and \begin{align*}x-1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 108?
  3. The product of two positive numbers is 120. Find the two numbers if one number is seven more than the other.
  4. Framing Warehouse offers a picture-framing service. The cost for framing a picture is made up of two parts. The cost of glass is $1 per square foot. The cost of the frame is $2 per linear foot. If the frame is a square, what size picture can you get framed for $20.00?

Mixed Review

  1. The area of a square varies directly with its side length.
    1. Write the general variation equation to model this sentence.
    2. If the area is 16 square feet when the side length is 4 feet, find the area when \begin{align*}s=1.5 \ feet\end{align*}.
  1. The surface area is the total amount of surface of a three-dimensional figure. The formula for the surface area of a cylinder is \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}, where \begin{align*}r=radius\end{align*} and \begin{align*}h=height \ of \ the \ cylinder\end{align*}. Determine the surface area of a soup can with a radius of 2 inches and a height of 5.5 inches.
  2. Factor \begin{align*}25g^2-36\end{align*}. Solve this polynomial when it equals zero.
  3. What is the greatest common factor of \begin{align*}343r^3 t, 21t^4\end{align*}, and \begin{align*}63rt^5\end{align*}?
  4. Discounts to the hockey game are given to groups with more than 12 people.
    1. Graph this solution on a number line.
    2. What is the domain of this situation?
    3. Will a church group with 12 members receive a discount?

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.11.

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Vocabulary

factoring completely

To factor a polynomial completely means to continue factoring until all factors other than monomial factors are prime factors.

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = 2^3 = 2 \times 2 \times 2 = 8.

factor

Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.

Factor to Solve

"Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of x that make each binomial equal to zero.

Quadratic Formula

The quadratic formula states that for any quadratic equation in the form ax^2+bx+c=0, x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.

Trinomial

A trinomial is a mathematical expression with three terms.

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