Factor by Grouping -3b(b-2)+7(b-2)^2
The question asks to apply the factor by grouping method to the given algebraic expression. Factor by grouping is a technique used in algebra to factor certain kinds of polynomials that have no common factors among all terms but can be rearranged into groups that do. The expression provided, -3b(b-2)+7(b-2)^2, appears to have a factor that is common to both terms which could potentially be factored out, following the rearrangement of the expression into separate groups if necessary.
$- 3 b \left(\right. b - 2 \left.\right) + 7 \left(\left(\right. b - 2 \left.\right)\right)^{2}$
Answer
Solution:
Step 1: Simplify each term.
Apply the distributive property to each term: $-3b(b - 2) + 7(b - 2)^2$.
Multiply $b$ by $b$ and $-2$ by $-3b$: $-3b^2 + 6b + 7(b - 2)^2$.
Expand the square $(b - 2)^2$: $-3b^2 + 6b + 7(b^2 - 4b + 4)$.
Distribute $7$ across $(b^2 - 4b + 4)$: $-3b^2 + 6b + 7b^2 - 28b + 28$.
Step 2: Combine like terms.
Add together $-3b^2$ and $7b^2$: $4b^2$.
Combine $6b$ and $-28b$: $-22b$.
The expression simplifies to: $4b^2 - 22b + 28$.
Step 3: Factor out the Greatest Common Factor (GCF).
The GCF of $4b^2$, $-22b$, and $28$ is $2$.
Factoring out $2$ gives: $2(2b^2 - 11b + 14)$.
Step 4: Factor by grouping.
For the trinomial $2b^2 - 11b + 14$, find two numbers that multiply to $2 \times 14 = 28$ and add to $-11$.
Rewrite $-11b$ as $-4b - 7b$: $2(2b^2 - 4b - 7b + 14)$.
Group terms and factor out the common factor from each group: $2((2b^2 - 4b) - (7b - 14))$.
Factor out $2b$ from the first group and $-7$ from the second: $2(2b(b - 2) - 7(b - 2))$.
Factor by grouping: $2(b - 2)(2b - 7)$.
Knowledge Notes:
Distributive Property: This property allows you to multiply a sum by multiplying each addend separately and then add the products. It is expressed as $a(b + c) = ab + ac$.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Greatest Common Factor (GCF): The largest factor that divides two or more numbers. When factoring expressions, the GCF is the largest expression that can be factored out of all terms in the polynomial.
Factoring by Grouping: A method used to factor polynomials with four or more terms. It involves grouping terms with common factors and factoring out the GCF from each group.
FOIL Method: A technique for expanding the product of two binomials. It stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.
Factoring Trinomials: A process for writing a trinomial as a product of two binomials. It often involves finding two numbers that multiply to give the product of the coefficient of the squared term and the constant term, and add to give the coefficient of the middle term.
