how would I write this in quadratic form?
4x to the eighth power plus 4x to the fourth power plus seven.. I am so confused..
To write the expression 4x^8 + 4x^4 + 7 in quadratic form, we need to rearrange it and group the terms with the same power of x together. Quadratic form typically represents an expression as a quadratic polynomial, which is a polynomial of degree 2.
Let's break down the given expression step-by-step:
1. Start with the given expression: 4x^8 + 4x^4 + 7
2. Notice that there are two terms with even powers of x (x^8 and x^4), while the constant term (7) doesn't have any x.
3. To begin grouping the terms, we can rewrite the expression as follows:
(4x^8 + 4x^4) + 7
4. Now, let's try to rewrite the grouped terms as a perfect square. Since both 4x^8 and 4x^4 have a common factor of 4, we can factor it out:
4(x^8 + x^4) + 7
5. Now, observe that the terms inside the parentheses, x^8 and x^4, share a common factor of x^4. We can factor it out as well:
4(x^4(x^4 + 1)) + 7
6. At this point, we can see that we have the sum of a perfect square (x^4 + 1) with a constant term (7). To complete the quadratic form, we need a quadratic polynomial in the parentheses. So, let's rewrite it by introducing a variable, let's say "y," which represents x^4:
4(y^2 + 1) + 7
7. Finally, we can expand the expression inside the parentheses to obtain the quadratic form:
4y^2 + 4 + 7
Simplifying further, we get:
4y^2 + 11
8. Since y represents x^4, we can substitute it back to obtain the final quadratic form:
4(x^4)^2 + 11
Which simplifies to:
4x^8 + 11
So, the quadratic form of the given expression 4x^8 + 4x^4 + 7 is 4x^8 + 11.