Factoring out the greatest common factor (GCF) is often the first step when factoring a polynomial. To find the GCF, identify the greatest numerical factor and the variables with the smallest exponents shared by every term. Then divide each term by the GCF and write the remaining expression inside parentheses.
Problems
Factor out the GCF
\(\textbf{1)}\) \( 15x^3 y+18x^4 y^5 \)
The answer is \( 3x^3 y(5+6xy^4) \)
Find the GCF of \(15x^3y\) and \(18x^4y^5\): \(3x^3y\).
\(15x^3y\div 3x^3y=5\)
\(18x^4y^5\div 3x^3y=6xy^4\)
Factor out the GCF: \(3x^3y(5+6xy^4)\).
Find the GCF of \(15x^3y\) and \(18x^4y^5\): \(3x^3y\).
\(15x^3y\div 3x^3y=5\)
\(18x^4y^5\div 3x^3y=6xy^4\)
Factor out the GCF: \(3x^3y(5+6xy^4)\).
\(\textbf{2)}\) \( 3x^2+18x^3 y^4 \)
The answer is \( 3x^2 (1+6xy^4) \)
The GCF of \(3x^2\) and \(18x^3y^4\) is \(3x^2\).
\(3x^2\div3x^2=1\)
\(18x^3y^4\div3x^2=6xy^4\)
Factor out the GCF: \(3x^2(1+6xy^4)\).
The GCF of \(3x^2\) and \(18x^3y^4\) is \(3x^2\).
\(3x^2\div3x^2=1\)
\(18x^3y^4\div3x^2=6xy^4\)
Factor out the GCF: \(3x^2(1+6xy^4)\).
\(\textbf{3)}\) \( 3xyz^2+x^2 y^2 z+9x^3 y \)
The answer is \( xy(3z^2+xyz+9x^2) \)
The variable factors common to all three terms are \(x\) and \(y\).
The GCF is \(xy\).
\(3xyz^2\div xy=3z^2\)
\(x^2y^2z\div xy=xyz\)
\(9x^3y\div xy=9x^2\)
Factor out the GCF: \(xy(3z^2+xyz+9x^2)\).
The variable factors common to all three terms are \(x\) and \(y\).
The GCF is \(xy\).
\(3xyz^2\div xy=3z^2\)
\(x^2y^2z\div xy=xyz\)
\(9x^3y\div xy=9x^2\)
Factor out the GCF: \(xy(3z^2+xyz+9x^2)\).
\(\textbf{4)}\) \( 5xyz+10wy \)
The answer is \( 5y(xz+2w) \)
The GCF of \(5xyz\) and \(10wy\) is \(5y\).
\(5xyz\div5y=xz\)
\(10wy\div5y=2w\)
Factor out the GCF: \(5y(xz+2w)\).
The GCF of \(5xyz\) and \(10wy\) is \(5y\).
\(5xyz\div5y=xz\)
\(10wy\div5y=2w\)
Factor out the GCF: \(5y(xz+2w)\).
\(\textbf{5)}\) \( a^2 d^2 y+a^2 d^3 y+ad^4 y^2 \)
The answer is \( ad^2 y(a+ad+d^2 y) \)
The variables shared by every term are \(a\), \(d^2\), and \(y\).
The GCF is \(ad^2y\).
\(a^2d^2y\div ad^2y=a\)
\(a^2d^3y\div ad^2y=ad\)
\(ad^4y^2\div ad^2y=d^2y\)
Factor out the GCF: \(ad^2y(a+ad+d^2y)\).
The variables shared by every term are \(a\), \(d^2\), and \(y\).
The GCF is \(ad^2y\).
\(a^2d^2y\div ad^2y=a\)
\(a^2d^3y\div ad^2y=ad\)
\(ad^4y^2\div ad^2y=d^2y\)
Factor out the GCF: \(ad^2y(a+ad+d^2y)\).
\(\textbf{6)}\) \( 5x^2 y+15x y^3 +10x^2y^2 \)
The answer is \( 5xy (x+3y^2+2xy) \)
The greatest common numerical factor is \(5\).
Every term also contains \(x\) and \(y\).
The GCF is \(5xy\).
\(5x^2y\div5xy=x\)
\(15xy^3\div5xy=3y^2\)
\(10x^2y^2\div5xy=2xy\)
Factor out the GCF: \(5xy(x+3y^2+2xy)\).
![Link to Youtube Video Solving Question Number 6](https://andymath.com/wp-content/uploads/2020/08/play-video.jpg")
The greatest common numerical factor is \(5\).
Every term also contains \(x\) and \(y\).
The GCF is \(5xy\).
\(5x^2y\div5xy=x\)
\(15xy^3\div5xy=3y^2\)
\(10x^2y^2\div5xy=2xy\)
Factor out the GCF: \(5xy(x+3y^2+2xy)\).
\(\textbf{7)}\) \( 2x^2 y +4x^3 y^2 +6x^4 y^3 \)
The answer is \( 2x^2 y\left(1+2xy+3x^2 y^2\right) \)
The GCF of the coefficients is \(2\).
The smallest powers shared by every term are \(x^2\) and \(y\).
The GCF is \(2x^2y\).
\(2x^2y\div2x^2y=1\)
\(4x^3y^2\div2x^2y=2xy\)
\(6x^4y^3\div2x^2y=3x^2y^2\)
Factor out the GCF: \(2x^2y(1+2xy+3x^2y^2)\).
The GCF of the coefficients is \(2\).
The smallest powers shared by every term are \(x^2\) and \(y\).
The GCF is \(2x^2y\).
\(2x^2y\div2x^2y=1\)
\(4x^3y^2\div2x^2y=2xy\)
\(6x^4y^3\div2x^2y=3x^2y^2\)
Factor out the GCF: \(2x^2y(1+2xy+3x^2y^2)\).
\(\textbf{8)}\) \( 4x^4+6x^3-8x^2 \)
The answer is \( 2x^2\left(2x^2+3x-4\right) \)
The GCF of \(4\), \(6\), and \(8\) is \(2\).
The smallest power of \(x\) is \(x^2\).
The GCF is \(2x^2\).
\(4x^4\div2x^2=2x^2\)
\(6x^3\div2x^2=3x\)
\(-8x^2\div2x^2=-4\)
Factor out the GCF: \(2x^2(2x^2+3x-4)\).
The GCF of \(4\), \(6\), and \(8\) is \(2\).
The smallest power of \(x\) is \(x^2\).
The GCF is \(2x^2\).
\(4x^4\div2x^2=2x^2\)
\(6x^3\div2x^2=3x\)
\(-8x^2\div2x^2=-4\)
Factor out the GCF: \(2x^2(2x^2+3x-4)\).
\(\textbf{9)}\) \( 12x^4-15x^2 \)
The answer is \( 3x^2\left(4x^2-5\right) \)
The GCF of \(12\) and \(15\) is \(3\).
Both terms contain \(x^2\).
The GCF is \(3x^2\).
\(12x^4\div3x^2=4x^2\)
\(-15x^2\div3x^2=-5\)
Factor out the GCF: \(3x^2(4x^2-5)\).
The GCF of \(12\) and \(15\) is \(3\).
Both terms contain \(x^2\).
The GCF is \(3x^2\).
\(12x^4\div3x^2=4x^2\)
\(-15x^2\div3x^2=-5\)
Factor out the GCF: \(3x^2(4x^2-5)\).
\(\textbf{10)}\) \( 12x^4-15x^3+3x^2 \)
The answer is \( 3x^2\left(4x^2-5x+1\right) \)
The GCF of the coefficients is \(3\).
The smallest power of \(x\) is \(x^2\).
The GCF is \(3x^2\).
\(12x^4\div3x^2=4x^2\)
\(-15x^3\div3x^2=-5x\)
\(3x^2\div3x^2=1\)
Factor out the GCF: \(3x^2(4x^2-5x+1)\).
The GCF of the coefficients is \(3\).
The smallest power of \(x\) is \(x^2\).
The GCF is \(3x^2\).
\(12x^4\div3x^2=4x^2\)
\(-15x^3\div3x^2=-5x\)
\(3x^2\div3x^2=1\)
Factor out the GCF: \(3x^2(4x^2-5x+1)\).
\(\textbf{11)}\) \( 14x^3+21x^2 \)
The answer is \(7x^2(2x+3)\)
The GCF of \(14\) and \(21\) is \(7\).
Both terms contain at least \(x^2\).
The GCF is \(7x^2\).
\(14x^3\div7x^2=2x\)
\(21x^2\div7x^2=3\)
Factor out the GCF: \(7x^2(2x+3)\).
The GCF of \(14\) and \(21\) is \(7\).
Both terms contain at least \(x^2\).
The GCF is \(7x^2\).
\(14x^3\div7x^2=2x\)
\(21x^2\div7x^2=3\)
Factor out the GCF: \(7x^2(2x+3)\).
\(\textbf{12)}\) \( 24a^4b^3-16a^2b^5 \)
The answer is \(8a^2b^3(3a^2-2b^2)\)
The GCF of \(24\) and \(16\) is \(8\).
The smallest powers shared are \(a^2\) and \(b^3\).
The GCF is \(8a^2b^3\).
\(24a^4b^3\div8a^2b^3=3a^2\)
\(-16a^2b^5\div8a^2b^3=-2b^2\)
Factor out the GCF: \(8a^2b^3(3a^2-2b^2)\).
The GCF of \(24\) and \(16\) is \(8\).
The smallest powers shared are \(a^2\) and \(b^3\).
The GCF is \(8a^2b^3\).
\(24a^4b^3\div8a^2b^3=3a^2\)
\(-16a^2b^5\div8a^2b^3=-2b^2\)
Factor out the GCF: \(8a^2b^3(3a^2-2b^2)\).
\(\textbf{13)}\) \( 18m^3n^2+30m^2n^4-12m^4n \)
The answer is \(6m^2n(3mn+5n^3-2m^2)\)
The GCF of \(18\), \(30\), and \(12\) is \(6\).
The smallest powers shared are \(m^2\) and \(n\).
The GCF is \(6m^2n\).
\(18m^3n^2\div6m^2n=3mn\)
\(30m^2n^4\div6m^2n=5n^3\)
\(-12m^4n\div6m^2n=-2m^2\)
Factor out the GCF: \(6m^2n(3mn+5n^3-2m^2)\).
The GCF of \(18\), \(30\), and \(12\) is \(6\).
The smallest powers shared are \(m^2\) and \(n\).
The GCF is \(6m^2n\).
\(18m^3n^2\div6m^2n=3mn\)
\(30m^2n^4\div6m^2n=5n^3\)
\(-12m^4n\div6m^2n=-2m^2\)
Factor out the GCF: \(6m^2n(3mn+5n^3-2m^2)\).
\(\textbf{14)}\) \( 9x^5-27x^3+36x^2 \)
The answer is \(9x^2(x^3-3x+4)\)
The GCF of \(9\), \(27\), and \(36\) is \(9\).
The smallest power of \(x\) is \(x^2\).
The GCF is \(9x^2\).
\(9x^5\div9x^2=x^3\)
\(-27x^3\div9x^2=-3x\)
\(36x^2\div9x^2=4\)
Factor out the GCF: \(9x^2(x^3-3x+4)\).
The GCF of \(9\), \(27\), and \(36\) is \(9\).
The smallest power of \(x\) is \(x^2\).
The GCF is \(9x^2\).
\(9x^5\div9x^2=x^3\)
\(-27x^3\div9x^2=-3x\)
\(36x^2\div9x^2=4\)
Factor out the GCF: \(9x^2(x^3-3x+4)\).
\(\textbf{15)}\) \( 16p^3q^2+40p^2q^3 \)
The answer is \(8p^2q^2(2p+5q)\)
The GCF of \(16\) and \(40\) is \(8\).
Both terms contain at least \(p^2q^2\).
The GCF is \(8p^2q^2\).
\(16p^3q^2\div8p^2q^2=2p\)
\(40p^2q^3\div8p^2q^2=5q\)
Factor out the GCF: \(8p^2q^2(2p+5q)\).
The GCF of \(16\) and \(40\) is \(8\).
Both terms contain at least \(p^2q^2\).
The GCF is \(8p^2q^2\).
\(16p^3q^2\div8p^2q^2=2p\)
\(40p^2q^3\div8p^2q^2=5q\)
Factor out the GCF: \(8p^2q^2(2p+5q)\).
\(\textbf{16)}\) \( -12x^3+20x^2-8x \)
The answer is \(-4x(3x^2-5x+2)\)
Because the leading term is negative, factor out a negative GCF.
The GCF of the coefficients is \(4\), and every term contains \(x\).
Factor out \(-4x\).
\(-12x^3\div(-4x)=3x^2\)
\(20x^2\div(-4x)=-5x\)
\(-8x\div(-4x)=2\)
The result is \(-4x(3x^2-5x+2)\).
Because the leading term is negative, factor out a negative GCF.
The GCF of the coefficients is \(4\), and every term contains \(x\).
Factor out \(-4x\).
\(-12x^3\div(-4x)=3x^2\)
\(20x^2\div(-4x)=-5x\)
\(-8x\div(-4x)=2\)
The result is \(-4x(3x^2-5x+2)\).
\(\textbf{17)}\) \( 35a^5b^2-49a^3b^4+14a^2b^3 \)
The answer is \(7a^2b^2(5a^3-7ab^2+2b)\)
The GCF of \(35\), \(49\), and \(14\) is \(7\).
The smallest powers shared are \(a^2\) and \(b^2\).
The GCF is \(7a^2b^2\).
\(35a^5b^2\div7a^2b^2=5a^3\)
\(-49a^3b^4\div7a^2b^2=-7ab^2\)
\(14a^2b^3\div7a^2b^2=2b\)
Factor out the GCF: \(7a^2b^2(5a^3-7ab^2+2b)\).
The GCF of \(35\), \(49\), and \(14\) is \(7\).
The smallest powers shared are \(a^2\) and \(b^2\).
The GCF is \(7a^2b^2\).
\(35a^5b^2\div7a^2b^2=5a^3\)
\(-49a^3b^4\div7a^2b^2=-7ab^2\)
\(14a^2b^3\div7a^2b^2=2b\)
Factor out the GCF: \(7a^2b^2(5a^3-7ab^2+2b)\).