Factor the expression completely: 60x^4 + 54x^3.
Understand the Problem
The question is asking to factor the algebraic expression completely, which involves identifying common factors in the terms of the expression and rewriting it in factored form.
Answer
The completely factored form is $6x^3(10x + 9)$.
Answer for screen readers
The completely factored form of the expression is: $$ 6x^3(10x + 9) $$
Steps to Solve
- Identify the Greatest Common Factor (GCF)
First, we need to find the GCF of the coefficients (60 and 54) and the variable terms.
The GCF of 60 and 54 can be calculated as:
- The prime factorization of 60: $60 = 2^2 \cdot 3 \cdot 5$
- The prime factorization of 54: $54 = 2 \cdot 3^3$
The common factors are $2$ and $3$, so the GCF is: $$ GCF = 2 \cdot 3 = 6 $$
For the variable terms, the lowest exponent of $x$ in $x^4$ and $x^3$ is $x^3$.
Thus, the GCF of the entire expression is: $$ GCF = 6x^3 $$
- Factor out the GCF
Next, we factor $6x^3$ out of each term in the expression: $$ 60x^4 + 54x^3 = 6x^3(10x + 9) $$
- Check the Result
To ensure our factoring is correct, we can distribute the $6x^3$ back into the parentheses:
- $6x^3 \cdot 10x = 60x^4$
- $6x^3 \cdot 9 = 54x^3$
Since this matches the original expression, the factoring is confirmed as correct.
The completely factored form of the expression is: $$ 6x^3(10x + 9) $$
More Information
Factoring is a key skill in algebra that involves removing common factors from expressions to simplify them. Identifying the GCF is a crucial first step in the factoring process and ensures that the expression can be rewritten in a simpler form.
Tips
- Overlooking the GCF: One common mistake is failing to find the GCF correctly. Always double-check the factorizations of coefficients and variable terms.
- Incorrectly distributing back: After factoring, some may not check by distributing back to the original expression, which can lead to errors.
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