Factor completely. 2z³ - 6z² + 5z - 15
Understand the Problem
The question is asking to factor the polynomial expression completely. The expression given is 2z³ - 6z² + 5z - 15.
Answer
The completely factored form of the expression is: $ (z - 3)(2z^2 + 5) $
Answer for screen readers
The completely factored form of the expression is:
$ (z - 3)(2z^2 + 5) $
Steps to Solve
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Write the expression The given polynomial expression is:
$$ 2z^3 - 6z^2 + 5z - 15 $$ -
Group the terms Group the first two terms and the last two terms:
$$ (2z^3 - 6z^2) + (5z - 15) $$ -
Factor out common factors from each group From the first group, factor out $2z^2$:
$$ 2z^2(z - 3) $$
From the second group, factor out $5$:
$$ 5(z - 3) $$
Now the expression looks like:
$$ 2z^2(z - 3) + 5(z - 3) $$
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Factor out the common binomial factor Notice that $(z - 3)$ is a common factor:
$$ (z - 3)(2z^2 + 5) $$ -
Write the final factored form The completely factored form of the expression is:
$$ (z - 3)(2z^2 + 5) $$
The completely factored form of the expression is:
$ (z - 3)(2z^2 + 5) $
More Information
In this polynomial, we factored by grouping, a common method used for polynomials of four terms. Factoring helps simplify expressions and is foundational in algebra.
Tips
- Forgetting to check for common factors in both groups.
- Not factoring correctly after grouping the terms.
- Assuming irreducibility too early without exploring all options.
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