7.4: Monomial Factors of Polynomials
Can you write the following polynomial as a product of a monomial and a polynomial?
\begin{align*}12x^4 + 6x^3 + 3x^2\end{align*}
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Khan Academy Factoring and the Distributive Property
Guidance
In the past you have studied common factors of two numbers. Consider the numbers 25 and 35. A common factor of 25 and 35 is 5 because 5 goes into both 25 and 35 evenly.
This idea can be extended to polynomials. A common factor of a polynomial is a number and/or variable that are a factor in all terms of the polynomial. The Greatest Common Factor (or GCF) is the largest monomial that is a factor of each of the terms of the polynomial.
To factor a polynomial means to write the polynomial as a product of other polynomials. One way to factor a polynomial is:
- Look for the greatest common factor.
- Write the polynomial as a product of the greatest common factor and the polynomial that results when you divide all the terms of the original polynomial by the greatest common factor.
One way to think about this type of factoring is that you are essentially doing the distributive property in reverse.
Example A
Factor the following binomial: \begin{align*}5a + 15\end{align*}
Solution: Step 1: Identify the GCF. Looking at each of the numbers, you can see that 5 and 15 can both be divided by 5. The GCF for this binomial is 5.
Step 2: Divide the GCF out of each term of the binomial:
\begin{align*}5a + 15 = 5(a + 3)\end{align*}
Example B
Factor the following polynomial: \begin{align*}4x^2+8x-2\end{align*}
Solution: Step 1: Identify the GCF. Looking at each of the numbers, you can see that 4, 8 and 2 can all be divided by 2. The GCF for this polynomial is 2.
Step 2: Divide the GCF out of each term of the polynomial:
\begin{align*}4x^2+8x-2=2(2x^2+4x-1)\end{align*}
Example C
Factor the following polynomial: \begin{align*}3x^5-9x^3-6x^2\end{align*}
Solution: Step 1: Identify the GCF. Looking at each of the terms, you can see that 3, 9 and 6 can all be divided by 3. Also notice that each of the terms has an \begin{align*}x^2\end{align*} in common. The GCF for this polynomial is \begin{align*}3x^2\end{align*}.
Step 2: Divide the GCF out of each term of the polynomial:
\begin{align*}3x^5-9x^3-6x^2=3x^2(x^3-3x-2)\end{align*}
Concept Problem Revisited
To write as a product you want to try to factor the polynomial: \begin{align*}12x^4 + 6x^3 + 3x^2\end{align*}.
Step 1: Identify the GCF of the polynomial. Looking at each of the numbers, you can see that 12, 6, and 3 can all be divided by 3. Also notice that each of the terms has an \begin{align*}x^2\end{align*} in common. The GCF for this polynomial is \begin{align*}3x^2\end{align*}.
Step 2: Divide the GCF out of each term of the polynomial:
\begin{align*}& 12x^4+6x^3+3x^2=3x^2(4x^2+2x+1)\end{align*}
Vocabulary
- Common Factor
- Common factors are numbers (numerical coefficients) or letters (literal coefficients) that are a factor in all parts of the polynomials.
- 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
- Find the common factors of the following: \begin{align*}a^2(b+7)-6(b+7)\end{align*}
- Factor the following polynomial: \begin{align*}5k^6+15k^4+10k^3+25k^2\end{align*}
- Factor the following polynomial: \begin{align*}27x^3y+18x^2y^2+9xy^3\end{align*}
Answers:
1. Step 1: Identify the GCF
- This problem is a little different in that if you look at the expression you notice that \begin{align*}(b + 7)\end{align*} is common in both terms. Therefore \begin{align*}(b + 7)\end{align*} is the common factor. The GCF for this expression is \begin{align*}(b + 7)\end{align*}.
- Step 2: Divide the GCF out of each term of the expression:
- \begin{align*}a^2 (b+7)-6(b+7)=(a^2-6)(b+7)\end{align*}
2. Step 1: Identify the GCF. Looking at each of the numbers, you can see that 5, 15, 10, and 25 can all be divided by 5. Also notice that each of the terms has an \begin{align*}k^2\end{align*} in common. The GCF for this polynomial is \begin{align*}5k^2\end{align*}.
- Step 2: Divide the GCF out of each term of the polynomial:
- \begin{align*}5k^6+15k^4+10k^3+25k^2=5k^2(k^4+3k^2+2k+5)\end{align*}
3. Step 1: Identify the GCF. Looking at each of the numbers, you can see that 27, 18 and 9 can all be divided by 9. Also notice that each of the terms has an \begin{align*}xy\end{align*} in common. The GCF for this polynomial is \begin{align*}9xy\end{align*}.
- Step 2: Divide the GCF out of each term of the polynomial:
- \begin{align*}27x^3y+18x^2y^2+9xy^3=9xy(3x^2+2xy+y^2)\end{align*}
Practice
Factor the following polynomials by looking for a common factor:
- \begin{align*}7x^2 + 14\end{align*}
- \begin{align*}9c^2+3\end{align*}
- \begin{align*}8a^2+4a\end{align*}
- \begin{align*}16x^2+24y^2\end{align*}
- \begin{align*}2x^2-12x+8\end{align*}
- \begin{align*}32w^2x+16xy+8x^2\end{align*}
- \begin{align*}12abc+6bcd+24acd\end{align*}
- \begin{align*}15x^2y-10x^2y^2+25x^2y\end{align*}
- \begin{align*}12a^2b-18ab^2-24a^2b^2\end{align*}
- \begin{align*}4s^3t^2-16s^2t^3+12st^2-24st^3\end{align*}
Find the common factors of the following expressions and then factor:
- \begin{align*}2x(x-5)+7(x-5)\end{align*}
- \begin{align*}4x(x-3)+5(x-3)\end{align*}
- \begin{align*}3x^2(e+4)-5(e+4)\end{align*}
- \begin{align*}8x^2(c-3)-7(c-3)\end{align*}
- \begin{align*}ax(x-b)+c(x-b)\end{align*}
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Greatest Common Factor
The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.Monomial
A monomial is an expression made up of only one term.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Image Attributions
Here you will learn to find a common factor in a polynomial and factor it out of the polynomial.