Factor 7z^4(2z^2-z-6)
The question provided is an algebraic factorization problem. You are asked to factor an expression, specifically a polynomial that is currently in a factored form involving a monomial term, 7z^4, multiplied by a trinomial, 2z^2 - z - 6. The goal is to break down the trinomial into two binomial expressions that when multiplied together give back the original trinomial, all while maintaining the monomial term as part of the expression. This requires finding two numbers that multiply to give the product of the leading coefficient and the constant term in the trinomial (i.e., 2 * -6 = -12) while also summing to the middle coefficient, which is -1 in this case. This technique is commonly referred to as the "AC method" or "splitting the middle term" in factorization.
$7 z^{4} \left(\right. 2 z^{2} - z - 6 \left.\right)$
Answer
Solution:
Step:1
Begin with the factorization by grouping method.
Step:1.1
For a quadratic expression $ax^2 + bx + c$, decompose the middle term into two terms whose product equals $a \cdot c = 2 \cdot -6 = -12$ and whose sum is $b = -1$.
Step:1.1.1
Extract $-1$ from $-z$ in the expression $7z^4(2z^2 - z - 6)$.
Step:1.1.2
Split $-1$ into the sum of $3$ and $-4$ resulting in $7z^4(2z^2 + (3 - 4)z - 6)$.
Step:1.1.3
Utilize the distributive property to separate the terms: $7z^4(2z^2 + 3z - 4z - 6)$.
Step:1.2
Identify and factor out the greatest common factor (GCF) from each term group.
Step:1.2.1
Pair the first two and the last two terms: $7z^4((2z^2 + 3z) - (4z + 6))$.
Step:1.2.2
Extract the GCF from each pair: $7z^4(z(2z + 3) - 2(2z + 3))$.
Step:1.3
Complete the factorization by taking out the common factor $2z + 3$: $7z^4((2z + 3)(z - 2))$.
Step:2
Simplify by removing extraneous parentheses: $7z^4(2z + 3)(z - 2)$.
Knowledge Notes:
Factor by Grouping: This is a method used to factor polynomials that have four terms. The process involves grouping terms with common factors and then factoring out these common factors.
Greatest Common Factor (GCF): The largest factor that divides two or more numbers. When factoring expressions, the GCF is the highest degree of any term that divides all other terms.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses. It is expressed as $a(b + c) = ab + ac$.
Quadratic Expressions: These are polynomials of degree two, generally in the form $ax^2 + bx + c$. Factoring such expressions often involves finding two numbers that multiply to $ac$ and add up to $b$.
Simplifying Expressions: This involves reducing expressions to their simplest form by removing parentheses and combining like terms where possible.
Factoring Polynomials: This is the process of breaking down a polynomial into simpler components (factors) that, when multiplied together, give the original polynomial. Factoring is a key step in solving polynomial equations.
