Algebra
Calculus
Trigonometry
Matrix
Question
Simplify the expression
x3−6x2+11x−63x−1
Evaluate
(x−1)(x−2)(x−3)3x−1
Solution
More Steps
Evaluate
(x−1)(x−2)(x−3)
Multiply the terms
More Steps
Evaluate
(x−1)(x−2)
Apply the distributive property
x×x−x×2−x−(−2)
Multiply the terms
x2−x×2−x−(−2)
Use the commutative property to reorder the terms
x2−2x−x−(−2)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
x2−2x−x+2
Subtract the terms
x2−3x+2
(x2−3x+2)(x−3)
Apply the distributive property
x2×x−x2×3−3x×x−(−3x×3)+2x−2×3
Multiply the terms
More Steps
Evaluate
x2×x
Use the product rule an×am=an+m to simplify the expression
x2+1
Add the numbers
x3
x3−x2×3−3x×x−(−3x×3)+2x−2×3
Use the commutative property to reorder the terms
x3−3x2−3x×x−(−3x×3)+2x−2×3
Multiply the terms
x3−3x2−3x2−(−3x×3)+2x−2×3
Multiply the numbers
x3−3x2−3x2−(−9x)+2x−2×3
Multiply the numbers
x3−3x2−3x2−(−9x)+2x−6
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
x3−3x2−3x2+9x+2x−6
Subtract the terms
More Steps
Evaluate
−3x2−3x2
Collect like terms by calculating the sum or difference of their coefficients
(−3−3)x2
Subtract the numbers
−6x2
x3−6x2+9x+2x−6
Add the terms
More Steps
Evaluate
9x+2x
Collect like terms by calculating the sum or difference of their coefficients
(9+2)x
Add the numbers
11x
x3−6x2+11x−6
x3−6x2+11x−63x−1
Show Solution
Find the excluded values
x=1,x=2,x=3
Evaluate
(x−1)(x−2)(x−3)3x−1
To find the excluded values,set the denominators equal to 0
(x−1)(x−2)(x−3)=0
Separate the equation into 3 possible cases
x−1=0x−2=0x−3=0
Solve the equation
More Steps
Evaluate
x−1=0
Move the constant to the right-hand side and change its sign
x=0+1
Removing 0 doesn't change the value,so remove it from the expression
x=1
x=1x−2=0x−3=0
Solve the equation
More Steps
Evaluate
x−2=0
Move the constant to the right-hand side and change its sign
x=0+2
Removing 0 doesn't change the value,so remove it from the expression
x=2
x=1x=2x−3=0
Solve the equation
More Steps
Evaluate
x−3=0
Move the constant to the right-hand side and change its sign
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
x=1x=2x=3
Solution
x=1,x=2,x=3
Show Solution
Rewrite the fraction
x−11−x−25+x−34
Evaluate
(x−1)(x−2)(x−3)3x−1
For each factor in the denominator,write a new fraction
x−1?+x−2?+x−3?
Write the terms in the numerator
x−1A+x−2B+x−3C
Set the sum of fractions equal to the original fraction
(x−1)(x−2)(x−3)3x−1=x−1A+x−2B+x−3C
Multiply both sides
(x−1)(x−2)(x−3)3x−1×(x−1)(x−2)(x−3)=x−1A×(x−1)(x−2)(x−3)+x−2B×(x−1)(x−2)(x−3)+x−3C×(x−1)(x−2)(x−3)
Simplify the expression
3x−1=(x2−5x+6)A+(x2−4x+3)B+(x2−3x+2)C
Simplify the expression
More Steps
Evaluate
(x2−5x+6)A+(x2−4x+3)B+(x2−3x+2)C
Multiply the terms
A(x2−5x+6)+(x2−4x+3)B+(x2−3x+2)C
Multiply the terms
A(x2−5x+6)+B(x2−4x+3)+(x2−3x+2)C
Multiply the terms
A(x2−5x+6)+B(x2−4x+3)+C(x2−3x+2)
Expand the expression
Ax2−5Ax+6A+B(x2−4x+3)+C(x2−3x+2)
Expand the expression
Ax2−5Ax+6A+Bx2−4Bx+3B+C(x2−3x+2)
Expand the expression
Ax2−5Ax+6A+Bx2−4Bx+3B+Cx2−3Cx+2C
3x−1=Ax2−5Ax+6A+Bx2−4Bx+3B+Cx2−3Cx+2C
Group the terms
3x−1=(A+B+C)x2+(−5A−4B−3C)x+6A+3B+2C
Equate the coefficients
⎩⎨⎧0=A+B+C3=−5A−4B−3C−1=6A+3B+2C
Swap the sides
⎩⎨⎧A+B+C=0−5A−4B−3C=36A+3B+2C=−1
Solve the equation for A
More Steps
Evaluate
A+B+C=0
Move the expression to the right-hand side and change its sign
A=0−(B+C)
Subtract the terms
A=−B−C
⎩⎨⎧A=−B−C−5A−4B−3C=36A+3B+2C=−1
Substitute the given value of A into the equation {−5A−4B−3C=36A+3B+2C=−1
{−5(−B−C)−4B−3C=36(−B−C)+3B+2C=−1
Simplify
{B+2C=36(−B−C)+3B+2C=−1
Simplify
{B+2C=3−3B−4C=−1
Solve the equation for B
{B=3−2C−3B−4C=−1
Substitute the given value of B into the equation −3B−4C=−1
−3(3−2C)−4C=−1
Simplify
More Steps
Evaluate
−3(3−2C)−4C
Expand the expression
−9+6C−4C
Subtract the terms
−9+2C
−9+2C=−1
Move the constant to the right-hand side and change its sign
2C=−1+9
Add the numbers
2C=8
Divide both sides
22C=28
Divide the numbers
C=28
Divide the numbers
More Steps
Evaluate
28
Reduce the numbers
14
Calculate
4
C=4
Substitute the given value of C into the equation B=3−2C
B=3−2×4
Calculate
B=−5
Substitute the given values of B,C into the equation A=−B−C
A=−(−5)−4
Simplify the expression
A=5−4
Calculate
A=1
Calculate
⎩⎨⎧A=1B=−5C=4
Solution
x−11−x−25+x−34
Show Solution
Find the roots
x=31
Alternative Form
x=0.3˙
Evaluate
(x−1)(x−2)(x−3)3x−1
To find the roots of the expression,set the expression equal to 0
(x−1)(x−2)(x−3)3x−1=0
Find the domain
More Steps
Evaluate
(x−1)(x−2)(x−3)=0
Apply the zero product property
⎩⎨⎧x−1=0x−2=0x−3=0
Solve the inequality
More Steps
Evaluate
x−1=0
Move the constant to the right side
x=0+1
Removing 0 doesn't change the value,so remove it from the expression
x=1
⎩⎨⎧x=1x−2=0x−3=0
Solve the inequality
More Steps
Evaluate
x−2=0
Move the constant to the right side
x=0+2
Removing 0 doesn't change the value,so remove it from the expression
x=2
⎩⎨⎧x=1x=2x−3=0
Solve the inequality
More Steps
Evaluate
x−3=0
Move the constant to the right side
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
⎩⎨⎧x=1x=2x=3
Find the intersection
x∈(−∞,1)∪(1,2)∪(2,3)∪(3,+∞)
(x−1)(x−2)(x−3)3x−1=0,x∈(−∞,1)∪(1,2)∪(2,3)∪(3,+∞)
Calculate
(x−1)(x−2)(x−3)3x−1=0
Cross multiply
3x−1=(x−1)(x−2)(x−3)×0
Simplify the equation
3x−1=0
Move the constant to the right side
3x=0+1
Removing 0 doesn't change the value,so remove it from the expression
3x=1
Divide both sides
33x=31
Divide the numbers
x=31
Check if the solution is in the defined range
x=31,x∈(−∞,1)∪(1,2)∪(2,3)∪(3,+∞)
Solution
x=31
Alternative Form
x=0.3˙
Show Solution