Math  /  Algebra

QuestionHow many roots, real or complex, does the polynomial 7+5x43x27+5 x^{4}-3 x^{2} have in all?
7 3 4 5

Studdy Solution

STEP 1

What is this asking? How many solutions, including complex numbers, does the equation 7+5x43x2=07 + 5x^4 - 3x^2 = 0 have? Watch out! Don't forget that complex solutions come in pairs, and don't mix up the highest power with the number of solutions!

STEP 2

1. Rewrite the polynomial
2. Apply the Fundamental Theorem of Algebra

STEP 3

Let's rewrite our polynomial 7+5x43x27 + 5x^4 - 3x^2 in descending order of powers of xx.
This gives us 5x43x2+75x^4 - 3x^2 + 7.
This makes it easier to see the **highest power** of xx, which is super important!

STEP 4

The **Fundamental Theorem of Algebra** says that a polynomial of degree *n* has *exactly* *n* roots, counting both real and complex roots.
Our polynomial 5x43x2+75x^4 - 3x^2 + 7 has a highest power of xx equal to **4**.
This is the **degree** of our polynomial.

STEP 5

So, our polynomial has **4** roots!
That's it!
Remember, some of these roots might be real, some might be complex, and some might even be repeated, but the total number of roots is always equal to the degree of the polynomial.

STEP 6

The polynomial has **4** roots.

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