Question

Factor out the greatest common factor in the expression.
5 x Superscript 4 Baseline minus 10 x cubed plus 35 x squared

Answer 1

To factor out the greatest common factor (GCF) from the expression \(5x^4 - 10x^3 + 35x^2\), we first identify the GCF of the coefficients and the variable part.

1. **Coefficients**: The coefficients are 5, -10, and 35. The GCF of these numbers is 5.
2. **Variable part**: The terms \(x^4\), \(x^3\), and \(x^2\) all have at least \(x^2\) as a common factor. The GCF for the variable \(x\) is \(x^2\).

Now we can combine these two GCFs:
\[
\text{GCF} = 5x^2
\]

Next, we factor \(5x^2\) out of each term in the expression:

\[
5x^4 \div 5x^2 = x^2 \quad \text{(first term)}
\]
\[
-10x^3 \div 5x^2 = -2x \quad \text{(second term)}
\]
\[
35x^2 \div 5x^2 = 7 \quad \text{(third term)}
\]

Thus, we can write the expression as:
\[
5x^2 \left( x^2 - 2x + 7 \right)
\]

So, the final factored form of the expression \(5x^4 - 10x^3 + 35x^2\) is:
\[
5x^2 (x^2 - 2x + 7)
\]