Certainly! Let's delve into the given polynomial expression [tex]\(8x^4 - 18x^3 + 20x^2 - 45x\)[/tex]. Below, I'll present the terms and break them down step-by-step.
### Step-by-Step Solution:
1. Identify the Polynomial:
The polynomial in question is:
[tex]\[
8x^4 - 18x^3 + 20x^2 - 45x
\][/tex]
2. Degree of the Polynomial:
The given polynomial is a 4th-degree polynomial. This is because the highest power of [tex]\( x \)[/tex] is 4.
3. Coefficients:
- The coefficient of [tex]\( x^4 \)[/tex] is 8.
- The coefficient of [tex]\( x^3 \)[/tex] is -18.
- The coefficient of [tex]\( x^2 \)[/tex] is 20.
- The coefficient of [tex]\( x \)[/tex] is -45.
4. Factor Out the GCD (Greatest Common Divisor):
To simplify the polynomial, we often start by factoring out the greatest common divisor (GCD) of all the terms. Here, each term is divisible by [tex]\( x \)[/tex], so we can factor out [tex]\( x \)[/tex]:
[tex]\[
8x^4 - 18x^3 + 20x^2 - 45x = x(8x^3 - 18x^2 + 20x - 45)
\][/tex]
Now we have factored out [tex]\( x \)[/tex], leaving us with the cubic polynomial [tex]\( 8x^3 - 18x^2 + 20x - 45 \)[/tex].
5. Further Analysis or Factorization:
To further simplify the cubic polynomial [tex]\( 8x^3 - 18x^2 + 20x - 45 \)[/tex], you would typically look for possible factorization methods such as factoring by grouping, finding rational roots, or using polynomial long division. However, since the roots and factorization are not immediately obvious and require several steps of algebraic manipulation, we can state our simplified term as:
[tex]\[
x(8x^3 - 18x^2 + 20x - 45)
\][/tex]
### Final Expression:
The factored form of the polynomial, with the common factor [tex]\( x \)[/tex] factored out, is:
[tex]\[
8x^4 - 18x^3 + 20x^2 - 45x = x(8x^3 - 18x^2 + 20x - 45)
\][/tex]
This gives a clear breakdown and understanding of the polynomial [tex]\( 8x^4 - 18x^3 + 20x^2 - 45x \)[/tex].
