How to Factor Polynomials by Taking a Common Factor?
A factor is a number that divides the given number without any remainder. It simply means expressing a number as a product of two other numbers. In this guide, you will learn more about factoring polynomials by taking a common factor.
Factor Polynomials by Taking a Common Factor: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Factor Polynomials by Taking a Common Factor: pop-up practice
The process of finding two or more expressions whose product is the given expression is known as the factorization of expressions.
Related Topics
A step-by-step guide to factoring polynomials by taking a common factor
The factor is a term used to express a number as a product of both numbers. Factorization is a way to find the factors of any mathematical object, whether it is a number, a polynomial, or an algebraic expression. Therefore, factorization of an algebraic expression refers to finding the factors of a given algebraic expression.
Follow the steps given below to find the factors of the expression by taking a common factor: \(x^2+4x\)
- Step 1: \(x^2\) can be factorized as \(x×x\), and \(4x\) can be factorized as \(x×4\).
- Step 2: Find the greatest common factor of these two terms. Here, we see that \(x\) is the largest common factor. Keep this coefficient out of parentheses, divide the polynomial expressions by this coefficient, and write the remaining expression inside the brackets.
- Step 3: Thus, the expression is factorized as \(x(x+4)\).
Factoring Polynomials by Taking a Common Factor – Example 1:
Factorize \(5z^3−15z^2\).
Solution:
The first term \(5z^3\) can be factorized as \(5×z×z×z\) and the second term \(15z^2\) can be factorized as \(15×z×z\).
The common factor for both terms is \(5z^2\).
Taking out the common factor, we get the factors \(5z^2(z−3)\).
Therefore, the factors of \(5z^3−15z^2\) are \(5z^2(z−3)\).
Exercises for Factoring Polynomials by Taking a Common Factor
Factorize the following expression by taking a common factor.
- \(\color{blue}{7x^2y+21xy^2}\)
- \(\color{blue}{6x^2y+14xy^2-42xy-2x^2y^2}\)
- \(\color{blue}{9x^5y^3+24x^3y^2-18x^2y}\)
- \(\color{blue}{32x^3y^3+56x^4y-40x^3y}\)
- \(\color{blue}{120x^3yz^3+32x^4y^2-68x^2y^2z}\)
- \(\color{blue}{7xy\left(x+3y\right)}\)
- \(\color{blue}{2xy\left(y-3\right)\left(-x+7\right)}\)
- \(\color{blue}{3x^2y\left(3x^3y^2+8xy-6\right)}\)
- \(\color{blue}{8x^3y\left(4y^2+7x-5\right)}\)
- \(\color{blue}{4x^2y\left(30xz^3+8x^2y-17yz\right)}\)
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