Examples
-1−1 , 00 , 11
Step 1
Roots are the points where the graph intercepts with the x-axis (y=0)(y=0).
y=0y=0 at the roots
Step 2
The root at x=-1x=−1 was found by solving for xx when x-(-1)=yx−(−1)=y and y=0y=0.
The factor is x+1x+1
Step 3
The root at x=0x=0 was found by solving for xx when x-(0)=yx−(0)=y and y=0y=0.
The factor is xx
Step 4
The root at x=1x=1 was found by solving for xx when x-(1)=yx−(1)=y and y=0y=0.
The factor is x-1x−1
Step 5
Combine all the factors into a single equation.
y=(x+1)(x)(x-1)y=(x+1)(x)(x−1)
Step 6
Step 6.1
Simplify by multiplying through.
Step 6.1.1
Apply the distributive property.
y=(x⋅x+1x)(x-1)y=(x⋅x+1x)(x−1)
Step 6.1.2
Simplify the expression.
Step 6.1.2.1
Multiply xx by xx.
y=(x2+1x)(x-1)y=(x2+1x)(x−1)
Step 6.1.2.2
Multiply xx by 11.
y=(x2+x)(x-1)y=(x2+x)(x−1)
y=(x2+x)(x-1)y=(x2+x)(x−1)
y=(x2+x)(x-1)y=(x2+x)(x−1)
Step 6.2
Expand (x2+x)(x-1)(x2+x)(x−1) using the FOIL Method.
Step 6.2.1
Apply the distributive property.
y=x2(x-1)+x(x-1)y=x2(x−1)+x(x−1)
Step 6.2.2
Apply the distributive property.
y=x2x+x2⋅-1+x(x-1)y=x2x+x2⋅−1+x(x−1)
Step 6.2.3
Apply the distributive property.
y=x2x+x2⋅-1+x⋅x+x⋅-1y=x2x+x2⋅−1+x⋅x+x⋅−1
y=x2x+x2⋅-1+x⋅x+x⋅-1y=x2x+x2⋅−1+x⋅x+x⋅−1
Step 6.3
Simplify and combine like terms.
Step 6.3.1
Simplify each term.
Step 6.3.1.1
Multiply x2x2 by xx by adding the exponents.
Step 6.3.1.1.1
Multiply x2x2 by xx.
Step 6.3.1.1.1.1
Raise xx to the power of 11.
y=x2x+x2⋅-1+x⋅x+x⋅-1y=x2x+x2⋅−1+x⋅x+x⋅−1
Step 6.3.1.1.1.2
Use the power rule aman=am+naman=am+n to combine exponents.
y=x2+1+x2⋅-1+x⋅x+x⋅-1y=x2+1+x2⋅−1+x⋅x+x⋅−1
y=x2+1+x2⋅-1+x⋅x+x⋅-1y=x2+1+x2⋅−1+x⋅x+x⋅−1
Step 6.3.1.1.2
Add 22 and 11.
y=x3+x2⋅-1+x⋅x+x⋅-1y=x3+x2⋅−1+x⋅x+x⋅−1
y=x3+x2⋅-1+x⋅x+x⋅-1y=x3+x2⋅−1+x⋅x+x⋅−1
Step 6.3.1.2
Move -1−1 to the left of x2x2.
y=x3-1⋅x2+x⋅x+x⋅-1y=x3−1⋅x2+x⋅x+x⋅−1
Step 6.3.1.3
Rewrite -1x2−1x2 as -x2−x2.
y=x3-x2+x⋅x+x⋅-1y=x3−x2+x⋅x+x⋅−1
Step 6.3.1.4
Multiply xx by xx.
y=x3-x2+x2+x⋅-1y=x3−x2+x2+x⋅−1
Step 6.3.1.5
Move -1−1 to the left of xx.
y=x3-x2+x2-1⋅xy=x3−x2+x2−1⋅x
Step 6.3.1.6
Rewrite -1x−1x as -x−x.
y=x3-x2+x2-xy=x3−x2+x2−x
y=x3-x2+x2-xy=x3−x2+x2−x
Step 6.3.2
Add -x2−x2 and x2x2.
y=x3+0-xy=x3+0−x
Step 6.3.3
Add x3x3 and 00.
y=x3-xy=x3−x
y=x3-xy=x3−x
y=x3-xy=x3−x
Step 7