What is the process for finding the greatest common factor of polynomials?
Understand the Problem
The content in the image provides a method for finding the greatest common factor (GCF) of polynomials. It outlines the steps and techniques, including organizing terms and identifying prime factors.
Answer
The GCF is \( x^2 \).
Answer for screen readers
The greatest common factor (GCF) of the polynomial ( x^4 + 7x^3 + 12x^2 ) is ( x^2 ).
Steps to Solve
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Identify the Polynomial Identify the given polynomial ( x^4 + 7x^3 + 12x^2 ).
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Factor Out Common Terms Factor out the greatest common factor (GCF) from the polynomial.
The terms are ( x^4, 7x^3, ) and ( 12x^2 ). The common factor is ( x^2 ).
So the expression becomes: $$ x^2(x^2 + 7x + 12) $$
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Factor the Quadratic Next, factor the quadratic ( x^2 + 7x + 12 ).
To factor, look for two numbers that multiply to ( 12 ) and add to ( 7 ). The numbers are ( 3 ) and ( 4 ).
Thus, we can write: $$ x^2 + 7x + 12 = (x + 3)(x + 4) $$
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Combine the Factors Now combine the factored terms to express the polynomial fully factored: $$ x^4 + 7x^3 + 12x^2 = x^2(x + 3)(x + 4) $$
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Identify the GCF The GCF of the original polynomial ( x^4 + 7x^3 + 12x^2 ) is: $$ x^2 $$
The greatest common factor (GCF) of the polynomial ( x^4 + 7x^3 + 12x^2 ) is ( x^2 ).
More Information
The GCF helps simplify polynomials and can be useful in solving equations or finding commonalities in expressions. Factoring polynomials is a crucial skill in algebra that assists in solving more complex mathematical problems.
Tips
- Forgetting to factor out the GCF before moving to further factorization.
- Not correctly identifying the terms that multiply to match the constant when factoring quadratics.
