Factor $2x^4 + 2x^3 - 20x^2$ completely.
Understand the Problem
The question asks us to factor the polynomial $2x^4 + 2x^3 - 20x^2$ completely. We should look for common factors first, and then factor the resulting quadratic expression if possible.
Answer
$2x^2(x^2 + x - 10)$
Answer for screen readers
$2x^2(x^2 + x - 10)$
Steps to Solve
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Find the greatest common factor (GCF) The GCF of the terms $2x^4$, $2x^3$, and $-20x^2$ is $2x^2$.
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Factor out the GCF Factoring out $2x^2$ from each term, we get: $2x^4 + 2x^3 - 20x^2 = 2x^2(x^2 + x - 10)$.
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Factor the quadratic expression (if possible) We need to check if the quadratic expression $x^2 + x - 10$ can be factored further. We look for two numbers that multiply to -10 and add to 1. The factors of -10 are: -1 and 10 -2 and 5 -5 and 2 -10 and 1 None of these pairs add up to 1. Therefore, the quadratic expression $x^2 + x - 10$ cannot be factored using integers.
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Write the final factored form Since the quadratic expression cannot be factored further, the completely factored form of the polynomial is $2x^2(x^2 + x - 10)$.
$2x^2(x^2 + x - 10)$
More Information
Factoring polynomials is a crucial skill in algebra. It allows us to simplify expressions, solve equations, and analyze functions. The greatest common factor (GCF) is always the first thing to look for when factoring a polynomial.
Tips
A common mistake is to stop after factoring out the GCF and not check if the resulting quadratic can be factored further. Another mistake is incorrectly factoring the quadratic expression.
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