Factoring by grouping is a strategy for factoring polynomials with four or more terms. The main idea is to split the expression into groups, factor out the greatest common factor from each group, and then factor out the common binomial. These problems practice basic grouping, rearranging terms before grouping, and factoring fully using grouping with other factoring patterns.
Notes
Practice Problems
Factor the following polynomials.
\(\textbf{1)}\) \( x^3+4x^2+3x+12 \)
The answer is \( (x^2+3)(x+4) \)
\(\,\,\,\,\,\,x^3+4x^2+3x+12\)
\(\,\,\,\,\,\,x^2\left(x+4\right)+3\left(x+4\right)\)
\(\,\,\,\,\,\,\left(x^2+3\right)\left(x+4\right)\)
\(\,\,\,\,\,\,x^3+4x^2+3x+12\)
\(\,\,\,\,\,\,x^2\left(x+4\right)+3\left(x+4\right)\)
\(\,\,\,\,\,\,\left(x^2+3\right)\left(x+4\right)\)
\(\textbf{2)}\) \( 8x^3-4x^2-6x+3 \)
The answer is \( \left(4x^2-3\right)(2x-1) \)
\(\,\,\,\,\,\,8x^3-4x^2-6x+3\)
\(\,\,\,\,\,\,4x^2\left(2x-1\right)-3\left(2x-1\right)\)
\(\,\,\,\,\,\,\left(4x^2-3\right)\left(2x-1\right)\)
\(\,\,\,\,\,\,8x^3-4x^2-6x+3\)
\(\,\,\,\,\,\,4x^2\left(2x-1\right)-3\left(2x-1\right)\)
\(\,\,\,\,\,\,\left(4x^2-3\right)\left(2x-1\right)\)
\(\textbf{3)}\) \(x^3+2x^2-5x-10\) The answer is \((x+2)\left(x^2-5\right)\)
\(\,\,\,\,\,\,x^3+2x^2-5x-10\)
\(\,\,\,\,\,\,x^2\left(x+2\right)-5\left(x+2\right)\)
\(\,\,\,\,\,\,\left(x^2-5\right)\left(x+2\right)\)
\(\,\,\,\,\,\,x^3+2x^2-5x-10\)
\(\,\,\,\,\,\,x^2\left(x+2\right)-5\left(x+2\right)\)
\(\,\,\,\,\,\,\left(x^2-5\right)\left(x+2\right)\)
\(\textbf{4)}\) \(3x^3+5x^2+6x+10\) The answer is \(\left(3x+5\right)\left(x^2+2\right)\)
\(\,\,\,\,\,\,3x^3+5x^2+6x+10\)
\(\,\,\,\,\,\,x^2(3x+5)+2(3x+5)\)
\(\,\,\,\,\,\,\left(3x+5\right)\left(x^2+2\right)\)
\(\,\,\,\,\,\,3x^3+5x^2+6x+10\)
\(\,\,\,\,\,\,x^2(3x+5)+2(3x+5)\)
\(\,\,\,\,\,\,\left(3x+5\right)\left(x^2+2\right)\)
\(\textbf{5)}\) \(3x^3+24x^2+2x+16\) The answer is \(\left(3x^2+2\right)\left(x+8\right)\)
\(\,\,\,\,\,\,3x^3+24x^2+2x+16\)
\(\,\,\,\,\,\,3x^2\left(x+8\right)+2\left(x+8\right)\)
\(\,\,\,\,\,\,\left(3x^2+2\right)\left(x+8\right)\)
\(\,\,\,\,\,\,3x^3+24x^2+2x+16\)
\(\,\,\,\,\,\,3x^2\left(x+8\right)+2\left(x+8\right)\)
\(\,\,\,\,\,\,\left(3x^2+2\right)\left(x+8\right)\)
\(\textbf{6)}\) \(11x-22-5x^3+10x^2\) The answer is \(\left(x-2\right)\left(-5x^2+11\right)\)
\(\,\,\,\,\,\,11x-22-5x^3+10x^2\)
\(\,\,\,\,\,\,11\left(x-2\right)-5x^2\left(x-2\right)\)
\(\,\,\,\,\,\,\left(11-5x^2\right)\left(x-2\right)\)
\(\,\,\,\,\,\,11x-22-5x^3+10x^2\)
\(\,\,\,\,\,\,11\left(x-2\right)-5x^2\left(x-2\right)\)
\(\,\,\,\,\,\,\left(11-5x^2\right)\left(x-2\right)\)
\(\textbf{7)}\) \(3x^3-15x^2+2x-10\) The answer is \(\left(3x^2+2\right)\left(x-5\right)\)
\(\,\,\,\,\,\,3x^3-15x^2+2x-10\)
\(\,\,\,\,\,\,3x^2\left(x-5\right)+2\left(x-5\right)\)
\(\,\,\,\,\,\,\left(3x^2+2\right)\left(x-5\right)\)
\(\,\,\,\,\,\,3x^3-15x^2+2x-10\)
\(\,\,\,\,\,\,3x^2\left(x-5\right)+2\left(x-5\right)\)
\(\,\,\,\,\,\,\left(3x^2+2\right)\left(x-5\right)\)
\(\textbf{8)}\) \(x^3+6x^2+4x+24\) The answer is \(\left(x^2+4\right)\left(x+6\right)\)
\(\,\,\,\,\,\,x^3+6x^2+4x+24\)
\(\,\,\,\,\,\,x^2\left(x+6\right)+4\left(x+6\right)\)
\(\,\,\,\,\,\,\left(x^2+4\right)\left(x+6\right)\)
\(\,\,\,\,\,\,x^3+6x^2+4x+24\)
\(\,\,\,\,\,\,x^2\left(x+6\right)+4\left(x+6\right)\)
\(\,\,\,\,\,\,\left(x^2+4\right)\left(x+6\right)\)
\(\textbf{9)}\) \(2x^3+7x^2+6x+21\) The answer is \(\left(2x+7\right)\left(x^2+3\right)\)
\(\,\,\,\,\,\,2x^3+7x^2+6x+21\)
\(\,\,\,\,\,\,x^2\left(2x+7\right)+3\left(2x+7\right)\)
\(\,\,\,\,\,\,\left(x^2+3\right)\left(2x+7\right)\)
\(\,\,\,\,\,\,2x^3+7x^2+6x+21\)
\(\,\,\,\,\,\,x^2\left(2x+7\right)+3\left(2x+7\right)\)
\(\,\,\,\,\,\,\left(x^2+3\right)\left(2x+7\right)\)
\(\textbf{10)}\) \(5x^3-15x^2+2x-6\) The answer is \(\left(5x^2+2\right)\left(x-3\right)\)
\(\,\,\,\,\,\,5x^3-15x^2+2x-6\)
\(\,\,\,\,\,\,5x^2\left(x-3\right)+2\left(x-3\right)\)
\(\,\,\,\,\,\,\left(5x^2+2\right)\left(x-3\right)\)
\(\,\,\,\,\,\,5x^3-15x^2+2x-6\)
\(\,\,\,\,\,\,5x^2\left(x-3\right)+2\left(x-3\right)\)
\(\,\,\,\,\,\,\left(5x^2+2\right)\left(x-3\right)\)
\(\textbf{11)}\) \(6x^3+9x^2+10x+15\) The answer is \(\left(3x+5\right)\left(2x^2+3\right)\)
\(\,\,\,\,\,\,6x^3+9x^2+10x+15\)
\(\,\,\,\,\,\,3x^2\left(2x+3\right)+5\left(2x+3\right)\)
\(\,\,\,\,\,\,\left(3x^2+5\right)\left(2x+3\right)\)
\(\,\,\,\,\,\,6x^3+9x^2+10x+15\)
\(\,\,\,\,\,\,3x^2\left(2x+3\right)+5\left(2x+3\right)\)
\(\,\,\,\,\,\,\left(3x^2+5\right)\left(2x+3\right)\)
\(\textbf{12)}\) \(4x^3-8x^2+3x-6\) The answer is \(\left(4x^2+3\right)\left(x-2\right)\)
\(\,\,\,\,\,\,4x^3-8x^2+3x-6\)
\(\,\,\,\,\,\,4x^2\left(x-2\right)+3\left(x-2\right)\)
\(\,\,\,\,\,\,\left(4x^2+3\right)\left(x-2\right)\)
\(\,\,\,\,\,\,4x^3-8x^2+3x-6\)
\(\,\,\,\,\,\,4x^2\left(x-2\right)+3\left(x-2\right)\)
\(\,\,\,\,\,\,\left(4x^2+3\right)\left(x-2\right)\)
\(\textbf{13)}\) \(xy+3x+2y+6\) The answer is \(\left(x+2\right)\left(y+3\right)\)
\(\,\,\,\,\,\,xy+3x+2y+6\)
\(\,\,\,\,\,\,x\left(y+3\right)+2\left(y+3\right)\)
\(\,\,\,\,\,\,\left(x+2\right)\left(y+3\right)\)
\(\,\,\,\,\,\,xy+3x+2y+6\)
\(\,\,\,\,\,\,x\left(y+3\right)+2\left(y+3\right)\)
\(\,\,\,\,\,\,\left(x+2\right)\left(y+3\right)\)
\(\textbf{14)}\) \(ab-4a+3b-12\) The answer is \(\left(a+3\right)\left(b-4\right)\)
\(\,\,\,\,\,\,ab-4a+3b-12\)
\(\,\,\,\,\,\,a\left(b-4\right)+3\left(b-4\right)\)
\(\,\,\,\,\,\,\left(a+3\right)\left(b-4\right)\)
\(\,\,\,\,\,\,ab-4a+3b-12\)
\(\,\,\,\,\,\,a\left(b-4\right)+3\left(b-4\right)\)
\(\,\,\,\,\,\,\left(a+3\right)\left(b-4\right)\)
\(\textbf{15)}\) \(12x^3+8x^2-9x-6\) The answer is \(\left(4x^2-3\right)\left(3x+2\right)\)
\(\,\,\,\,\,\,12x^3+8x^2-9x-6\)
\(\,\,\,\,\,\,4x^2\left(3x+2\right)-3\left(3x+2\right)\)
\(\,\,\,\,\,\,\left(4x^2-3\right)\left(3x+2\right)\)
\(\,\,\,\,\,\,12x^3+8x^2-9x-6\)
\(\,\,\,\,\,\,4x^2\left(3x+2\right)-3\left(3x+2\right)\)
\(\,\,\,\,\,\,\left(4x^2-3\right)\left(3x+2\right)\)
Challenge Problems
Factor fully.
\(\textbf{16)}\) \( x^5-4x^3+x^2-4 \)
The answer is \( (x+1)(x^2-x+1)(x+2)(x-2) \)
\(\,\,\,\,\,\,x^5-4x^3+x^2-4\)
\(\,\,\,\,\,\,x^3\left(x^2-4\right)+1\left(x^2-4\right)\)
\(\,\,\,\,\,\,\left(x^3+1\right)\left(x^2-4\right)\)
\(\,\,\,\,\,\,\text{Use }a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)
\(\,\,\,\,\,\,\left(x+1\right)\left(x^2-x+1\right)\left(x^2-4\right)\)
\(\,\,\,\,\,\,\text{Use }a^2-b^2=\left(a+b\right)\left(a-b\right)\)
\(\,\,\,\,\,\,\left(x+1\right)\left(x^2-x+1\right)\left(x+2\right)\left(x-2\right)\)
\(\,\,\,\,\,\,x^5-4x^3+x^2-4\)
\(\,\,\,\,\,\,x^3\left(x^2-4\right)+1\left(x^2-4\right)\)
\(\,\,\,\,\,\,\left(x^3+1\right)\left(x^2-4\right)\)
\(\,\,\,\,\,\,\text{Use }a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)
\(\,\,\,\,\,\,\left(x+1\right)\left(x^2-x+1\right)\left(x^2-4\right)\)
\(\,\,\,\,\,\,\text{Use }a^2-b^2=\left(a+b\right)\left(a-b\right)\)
\(\,\,\,\,\,\,\left(x+1\right)\left(x^2-x+1\right)\left(x+2\right)\left(x-2\right)\)
\(\textbf{17)}\) \( 48xy-3xz+80wy-5wz \)
The answer is \( (3x+5w)(16y-z) \)
\(\,\,\,\,\,\,48xy-3xz+80wy-5wz\)
\(\,\,\,\,\,\,3x\left(16y-z\right)+5w\left(16y-z\right)\)
\(\,\,\,\,\,\,\left(3x+5w\right)\left(16y-z\right)\)
\(\,\,\,\,\,\,48xy-3xz+80wy-5wz\)
\(\,\,\,\,\,\,3x\left(16y-z\right)+5w\left(16y-z\right)\)
\(\,\,\,\,\,\,\left(3x+5w\right)\left(16y-z\right)\)
\(\textbf{18)}\) \(5x+5x^3+2x^4+2x^6\) The answer is \( \left(x\right)\left(x^2+1\right)\left(2x^3+5\right)\)
\(\,\,\,\,\,\,5x+5x^3+2x^4+2x^6\)
\(\,\,\,\,\,\,x\left(5+5x^2+2x^3+2x^5\right)\)
\(\,\,\,\,\,\,x\left(5\left(1+x^2\right)+2x^3\left(1+x^2\right)\right)\)
\(\,\,\,\,\,\,x\left(5+2x^3\right)\left(1+x^2\right)\)
\(\,\,\,\,\,\,x\left(x^2+1\right)\left(2x^3+5\right)\)
\(\,\,\,\,\,\,5x+5x^3+2x^4+2x^6\)
\(\,\,\,\,\,\,x\left(5+5x^2+2x^3+2x^5\right)\)
\(\,\,\,\,\,\,x\left(5\left(1+x^2\right)+2x^3\left(1+x^2\right)\right)\)
\(\,\,\,\,\,\,x\left(5+2x^3\right)\left(1+x^2\right)\)
\(\,\,\,\,\,\,x\left(x^2+1\right)\left(2x^3+5\right)\)
\(\textbf{19)}\) \(2x^7-8x^5-16x^4+64x^2\) The answer is \(\,\,\,2x^2\left(x-2\right)^2\left(x+2\right)\left(x^2+2x+4\right)\)
\(\,\,\,\,\,\,2x^7-8x^5-16x^4+64x^2\)
\(\,\,\,\,\,\,2x^2\left(x^5-4x^3-8x^2+32\right)\)
\(\,\,\,\,\,\,2x^2\left(x^3(x^2-4)-8(x^2-4)\right)\)
\(\,\,\,\,\,\,2x^2\left(x^2-4\right)\left(x^3-8\right)\)
\(\,\,\,\,\,\,2x^2\left(x-2\right)\left(x+2\right)\left(x-2\right)\left(x^2+2x+4\right)\)
\(\,\,\,\,\,\,2x^2\left(x-2\right)^2\left(x+2\right)\left(x^2+2x+4\right)\)
\(\,\,\,\,\,\,2x^7-8x^5-16x^4+64x^2\)
\(\,\,\,\,\,\,2x^2\left(x^5-4x^3-8x^2+32\right)\)
\(\,\,\,\,\,\,2x^2\left(x^3(x^2-4)-8(x^2-4)\right)\)
\(\,\,\,\,\,\,2x^2\left(x^2-4\right)\left(x^3-8\right)\)
\(\,\,\,\,\,\,2x^2\left(x-2\right)\left(x+2\right)\left(x-2\right)\left(x^2+2x+4\right)\)
\(\,\,\,\,\,\,2x^2\left(x-2\right)^2\left(x+2\right)\left(x^2+2x+4\right)\)
\(\textbf{20)}\) \(x^4+2x^3-9x^2-18x\) The answer is \(x\left(x+2\right)\left(x+3\right)\left(x-3\right)\)
\(\,\,\,\,\,\,x^4+2x^3-9x^2-18x\)
\(\,\,\,\,\,\,x\left(x^3+2x^2-9x-18\right)\)
\(\,\,\,\,\,\,x\left(x^2\left(x+2\right)-9\left(x+2\right)\right)\)
\(\,\,\,\,\,\,x\left(x^2-9\right)\left(x+2\right)\)
\(\,\,\,\,\,\,x\left(x+3\right)\left(x-3\right)\left(x+2\right)\)
\(\,\,\,\,\,\,x^4+2x^3-9x^2-18x\)
\(\,\,\,\,\,\,x\left(x^3+2x^2-9x-18\right)\)
\(\,\,\,\,\,\,x\left(x^2\left(x+2\right)-9\left(x+2\right)\right)\)
\(\,\,\,\,\,\,x\left(x^2-9\right)\left(x+2\right)\)
\(\,\,\,\,\,\,x\left(x+3\right)\left(x-3\right)\left(x+2\right)\)
See Related Pages\(\)
\(\bullet\text{ Factoring Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Factoring out a GCF}\)
\(\,\,\,\,\,\,\,\,3xyz^2+x^2y^2z+9x^3y=xy(3z^2+xyz+9x^2)…\)
\(\bullet\text{ Perfect Square Trinomials}\)
\(\,\,\,\,\,\,\,\,x^2-6x+9=(x-3)^2…\)
\(\bullet\text{ Factoring Trinomials with a}=1\)
\(\,\,\,\,\,\,\,\,x^2+7x+12=(x+3)(x+4)…\)
\(\bullet\text{ Factoring Trinomials with a} \ne 1\)
\(\,\,\,\,\,\,\,\,3x^2+11x+6=(3x+2)(x+3)…\)
\(\bullet\text{ Factoring with u-substitution}\)
\(\,\,\,\,\,\,\,\,x^4+5x^2+6=u^2+5u+6…\)
\(\bullet\text{ Difference of Two Squares}\)
\(\,\,\,\,\,\,\,\,x^2-16=(x+4)(x-4)…\)
\(\bullet\text{ Sum/Difference of Two Cubes}\)
\(\,\,\,\,\,\,\,\,x^3-8=(x-2)(x^2+2x+4)…\)
\(\bullet\text{ Solving Quadratic Equations by Factoring}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)