RTCNTC said:Precalculus by David Cohen, 3rd Edition
Chapter 1, Section 1.3.
Question 54.
Factor the expression.
27 - (a - b)^3
Must I expand (a - b)^3 or is this expression the difference of cubes?
RTCNTC said:27 - (a - b)^3
3^3 - (a - b)^3
The difference of cubes formula:
a^3 – b^3 = (a – b)(a^2 + ab + b^2)
I will let a = 3 and b = (a - b).
(3 - (a - b))(3^2 + 3(a - b) + (a - b)^2)
(3 - a + b)(9 + 3a - 3b + (a - b)(a + b))
(3 - a + b)(9 + 3a - 3b + a^2 -- b^2)
I cannot simplify anymore.
kaliprasad said:there are 2 issues here
a = 3 and b = (a - b) is bad notation .your are using a for 2 different purposes. you could use $x^3-y^3$ with $x=3$ and $y = a-b$
secondly $(a-b)^2$ you have changed to $(a-b)(a+b)$
RTCNTC said:In that case, can you work it out for me step by step?
kaliprasad said:your steps are correct till you instead of expanding $(a-b)^2$ did $(a-b)(a+b) $ and I work it out here
$27-(a-b)^3$
= $3^3-(a-b)^3$
$= (3-(a-b))(3 + (a-b) + (a-b)^2)$
MarkFL said:This last step should be:
\(\displaystyle \left(3-(a-b)\right)\left(3^2+3(a-b)+(a-b)^2\right)\) :D
MarkFL said:This last step should be:
\(\displaystyle \left(3-(a-b)\right)\left(3^2+3(a-b)+(a-b)^2\right)\) :D
Factor 27 - (a - b)^3 is a mathematical expression that represents a polynomial. It is a combination of numbers, variables, and exponents that can be simplified and solved using algebraic methods.
To factor 27 - (a - b)^3, you can use the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, a = 3 and b = (a - b) = (a - b)^2.
Yes, the expression can be simplified further by expanding the brackets and combining like terms. The final simplified form would be 27 - (a^3 - 3a^2b + 3ab^2 - b^3) = 27 - a^3 + 3a^2b - 3ab^2 + b^3.
Yes, there is a specific order of operations that should be followed when solving this expression. First, simplify any exponents. Then, use the distributive property to expand the brackets. Finally, combine like terms and simplify the expression.
The purpose of factoring 27 - (a - b)^3 is to break down a complex expression into simpler terms. This can make it easier to solve the expression, identify any patterns or relationships, and find the solutions or roots of the equation.