Factor by Grouping (x^2-6x)^2+(x^2-6x)-56
The given problem asks to use the method of factor by grouping on the algebraic expression provided. Factor by grouping is a technique for factoring polynomials that involves rearranging the terms of the polynomial into groups that have a common factor. The expression given is a polynomial that appears to be structured in a way that might allow for it to be factored through this method, with the goal of simplifying the expression into a product of simpler polynomials or factors. The terms within the brackets, (x^2-6x), appear more than once, indicating that the expression may be reorganized for grouping. The problem is asking to apply this technique to see if the expression can be factorially simplified.
$\left(\left(\right. x^{2} - 6 x \left.\right)\right)^{2} + \left(\right. x^{2} - 6 x \left.\right) - 56$
Answer
Solution:
Step 1
The given polynomial is not suitable for factoring by grouping. We should consider an alternative factoring strategy or, if uncertain, proceed with a general factoring approach.
Step 2
Express $(x^2 - 6x)^2$ as $(x^2 - 6x)(x^2 - 6x)$.
Thus, the polynomial becomes $(x^2 - 6x)(x^2 - 6x) + x^2 - 6x - 56$.
Step 3
Expand $(x^2 - 6x)(x^2 - 6x)$ using the FOIL method.
Step 3.1
Distribute $x^2(x^2 - 6x) - 6x(x^2 - 6x) + x^2 - 6x - 56$.
Step 3.2
Distribute $x^4 - x^2(6x) - 6x(x^2) + 6x(6x) + x^2 - 6x - 56$.
Step 3.3
Continue with the distribution to get $x^4 - 6x^3 - 6x^3 + 36x^2 + x^2 - 6x - 56$.
Step 4
Combine like terms and simplify.
Step 4.1
Simplify each term individually.
Step 4.1.1
Multiply $x^2$ by $x^2$ to get $x^{2+2}$.
Step 4.1.2
Use the commutative property to reorder the terms: $x^4 - 6x^3 - 6x^3 + 36x^2 + x^2 - 6x - 56$.
Step 4.1.3
Combine the exponents where applicable.
Step 4.1.4
Simplify the multiplication of $x$ and $x^2$ to get $x^{2+1}$.
Step 4.1.5
Apply the commutative property again if necessary.
Step 4.1.6
Combine the exponents for $x \cdot x$ to get $x^{1+1}$.
Step 4.1.7
Multiply $-6$ by $-6$ to get $+36$.
Step 4.2
Combine the like terms $-6x^3$ and $-6x^3$ to get $-12x^3$.
Step 5
Add $36x^2$ and $x^2$ to get $37x^2$.
The final expression is $x^4 - 12x^3 + 37x^2 - 6x - 56$.
Knowledge Notes:
To solve the given problem, we need to understand several algebraic concepts and properties:
Factor by Grouping: A method used to factor polynomials that involves grouping terms with common factors and then factoring out the greatest common factor from each group.
FOIL Method: A technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms in the binomials.
Distributive Property: This property states that $a(b + c) = ab + ac$. It allows us to multiply a term across the terms within a parenthesis.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product, i.e., $ab = ba$.
Combining Like Terms: This involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power.
Power Rule for Exponents: When multiplying like bases, you add the exponents, as in $a^m \cdot a^n = a^{m+n}$.
Simplifying Expressions: This process involves using the above properties and rules to rewrite expressions in a simpler or more compact form.
By applying these concepts and properties systematically, we can simplify the given polynomial expression to its factored or expanded form.
