How can I express the polynomial 6x^54 + 4x^52 + 6 in the form of a quadratic using substitution?

To rewrite the polynomial 6x⁵⁴ + 4x⁵² + 6 in a quadratic form, we can use substitution. Let’s choose a substitution that simplifies the expression significantly. A good substitution in this case is to let:

y = x²⁶

This choice is effective because:

• x⁵⁴ = (x²⁶)² * x² = (y²) * x²
• x⁵² = (x²⁶)² = (y)²

Next, rewrite the original polynomial using this substitution:

6x⁵⁴ + 4x⁵² + 6 = 6(y² * x²) + 4y + 6

Notice that the term with the highest degree, 6x⁵⁴, becomes part of a quadratic where the variable is y. So, if we consider only the terms in the variable y and gather the x terms, we simplify further:

6x²y² + 4y + 6

Now, this doesn’t fit perfectly into the form of a quadratic equation in y alone without considering x. The final polynomial can be expressed as:

6x²y² + 4y + 6 where y = x²⁶.

For quadratic equations, we usually want something simpler without additional variables like x. If we treat x as a constant while converting y terms, we can gauge the quadratic expression better.